Diffusive wave in the low Mach limit for compressible Navier–Stokes equations

Advances in Mathematics 319 (2017) 348–395

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Advances in Mathematics www.elsevier.com/locate/aim

Diffusive wave in the low Mach limit for compressible Navier–Stokes equations Feimin Huang a , Tian-Yi Wang b,c,∗ , Yong Wang a a

Institute of Applied Mathematics, AMSS, CAS, Beijing 100190, China Department of Mathematics, School of Science, Wuhan University of Technology, Wuhan, China c Gran Sasso Science Institute, viale Francesco Crispi, 7, 67100 L’Aquila, Italy b

a r t i c l e

i n f o

Article history: Received 16 April 2016 Received in revised form 27 October 2016 Accepted 1 August 2017 Available online xxxx Communicated by Camillo De Lellis MSC: 35Q35 35B65 76N10 Keywords: Compressible Navier–Stokes equations Low Mach limit Nonlinear diffusion wave Well-prepared data Ill-prepared data

a b s t r a c t The low Mach limit for 1D non-isentropic compressible Navier–Stokes flow, whose density and temperature have different asymptotic states at infinity, is rigorously justified. The problems are considered on both well-prepared and illprepared data. For the well-prepared data, the solutions of compressible Navier–Stokes equations are shown to converge to a nonlinear diffusion wave solution globally in time as Mach number goes to zero when the difference between the states at ±∞ is suitably small. In particular, the velocity of diffusion wave is only driven by the variation of temperature. It is further shown that the solution of compressible Navier–Stokes system also has the same property when Mach number is small, which has never been observed before. The convergence rates on both Mach number and time are also obtained for the well-prepared data. For the ill-prepared data, the limit relies on the uniform estimates including weighted time derivatives and an extended convergence lemma. And the difference between the states at ±∞ can be arbitrary large in this case. © 2017 Elsevier Inc. All rights reserved.

* Corresponding author. E-mail addresses: [email protected] (F. Huang), [email protected] (T.-Y. Wang), [email protected] (Y. Wang). http://dx.doi.org/10.1016/j.aim.2017.08.004 0001-8708/© 2017 Elsevier Inc. All rights reserved.

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Contents 1.

Introduction . . . . . . . . . . . . . . . . . . . . . 1.1. Main results for well-prepared data . 1.2. Main results for ill-prepared data . . 2. Well-prepared data . . . . . . . . . . . . . . . . . 3. Ill-prepared data . . . . . . . . . . . . . . . . . . 3.1. Uniform estimates . . . . . . . . . . . . 3.2. Low Mach limit . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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349 353 356 359 373 373 389 394 394

1. Introduction The non-isentropic Navier–Stokes system in Rn is as follows ⎧ ⎪ ⎪ ⎪∂t ρ + div(ρu) = 0, ⎨

∂t (ρu) + div(ρu ⊗ u) + ∇P = div(2μD(u)) + ∇(λdivu), ⎪ ⎪ ⎪ ⎩∂ (ρ(e + 1 |u|2 )) + div(ρu(e + 1 |u|2 ) + P u) = div(κ∇T ) + div(2μD(u)u + λdivuu), t 2 2 (1.1) for t > 0, x ∈ Rn . Here the unknown functions ρ, u, and T represent the density, velocity, and temperature, respectively. The pressure function and internal function are defined by P = RρT , e = cv T ,

(1.2)

where the parameters R > 0 and cv > 0 are the gas constant and the heat capacity at constant volume, respectively. D(u) is the deformation tensor given by D(u) =

1 (∇u + (∇u)t ), 2

where (∇u)t denote the transpose of matrix ∇u. μ and λ are the Lamé viscosity coefficients which satisfy μ > 0, 2μ + nλ > 0, and κ > 0 is the heat conductivity coefficient. For simplicity, we assume that μ, λ, κ are constants. It is noted cv = (γ − 1)/R where γ > 1 is the adiabatic exponent. In this paper, since we consider the low Mach limit for smooth solutions of non-isentropic compressible Navier–Stokes equation, we normalize R = 1 and cv = 1 for simplicity of presentation, see also [25]. And we point out that our results and energy estimates in this paper still

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hold for any given cv > 0 (equivalently for any given γ > 1), the only difference is that some constants of this paper may depend on cv . Let ε be the compressibility parameter which is a nondimensional quantity. As in [39], we set t → εt, x → x, u → εu, μ → εμ, λ → ελ, κ → εκ.

(1.3)

Based on the above changes of variables, the compressible Navier–Stokes system (1.1), written after the nondimensionalization, becomes ⎧ ε ε ε ⎪ ⎪ ⎪∂t ρ + div(ρ u ) = 0, ⎨

ε ε (1.4) ∂t (ρε uε ) + div(ρε uε ⊗ uε ) + ∇P ε2 = div(2μD(u )) + ∇(λdivu ), ⎪ ⎪ ⎪ ⎩∂ (ρε T ε ) + div(ρε uε T ε ) + P ε divuε = div(κ∇T ε ) + ε2 [2μ|D(uε )|2 + λ|divuε |2 ], t ε

in which the typical mean fluid velocity has been chosen as the ratio of time units to space units. Accordingly, the parameter ε essentially presents the maximum Mach number of the fluid. The limit of solutions of (1.4) as ε goes to 0 is usually called as low Mach limit [29,30]. The purpose of the low Mach number approximation is to justify that the compression, due to pressure variations, can be neglected. This is a common assumption when discussing the fluid dynamics of highly subsonic flows. Similar to [2,39], we assume that the pressure is a small perturbation of a given constant state P > 0, i.e. P ε = P + O(ε),

(1.5)

which satisfies the following equation ∂t (P ε ) + div(uε P ε ) + P ε divuε = div(κ∇T ε ) + ε2 [2μ|D(uε )|2 + λ|divuε |2 ].

(1.6)

Formally, as the Mach number tends to zero, the equations (1.6), (1.4)2 and (1.4)3 become ⎧ ⎪ ⎪ ⎪div(2P u − κ∇T ) = 0, ⎨

ρ(ut + u · ∇u) + ∇π = div(2μD(u)) + ∇(λdivu), ⎪ ⎪ ⎪ ⎩ρ(T + u · ∇T ) + P divu = div(κ∇T ),

(1.7)

t

and the density satisfies ρ = P /T and ρt + div(ρu) = 0.

(1.8)

Without loss of generality, we normalize P to be 1. Let the approximate initial data ε (pεin , uεin , Tin ) converge to (pin , uin , Tin ) as ε → 0 in some sense. Then, the approximate

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initial data is regarded as well-prepared data if uin and Tin satisfies (1.7)1 . Otherwise, it is called as ill-prepared data. The low Mach limit is an important and interesting problem in the fluid dynamics. There have been a lot of literatures on the low Mach limit. The first result is due to Klainerman and Majda [29,30], in which they proved the incompressible limit of the isentropic Euler equations to the incompressible Euler equations for local smooth solutions with well-prepared data. By using the fast decay of acoustic waves, Ukai [42] verified the low Mach limit for the general data, see also [3]. The half plane and exterior domain cases were considered in [19–21]. In [37], Schochet showed the limit of the full compressible Euler equations to the incompressible inhomogeneous Euler equations in bounded domain for local smooth solutions with well-prepared initial data. He [38] further studied the fast singular limit for general hyperbolic partial differential equations. The major breakthrough on the ill-prepared data is due to Métivier and Schochet [35], in which they proved the low Mach limit of full Euler equations in the whole space by a significant convergence lemma on acoustic waves. Later, Alazard [1] extended [35] to the exterior domain. The one dimension spatial periodic case on the full compressible Euler equations was considered in [36]. For other interesting works, see [6] and the references therein. For the isentropic Navier–Stokes equations, the low Mach limit of the global weak solutions with general initial data have been well studied under various boundary conditions, see [9,10,32,33]. For the non-isentropic Navier–Stokes equations, Alazard [2] justified the low Mach limit in the whole space for the ill-prepared data, by employing a uniform estimate and the convergence lemma of [35]. For the bounded domain, the low Mach limit was justified by Jiang–Ou [26] and Dou–Jiang–Ou [11]. A dispersive Navier–Stokes system was also studied in [31]. For other interesting works, see [7,8,14,27,34,39] for Navier–Stokes equations, [13,15,22–25] for MHD equations and the references therein. Note that in the whole space, all results above require that both density and temperature have the constant background, i.e. ρε (x, t) → ρ, and T ε (x, t) → T , as |x| → ∞,

(1.9)

where ρ > 0 and T > 0 are given constants. In this paper, we will study the low Mach limit when the background is not constant state in the one dimensional case, that is (ρε , T ε )(x, t) → (ρ± , T± ), as x → ±∞, with ρ− T− = ρ+ T+ ,

(1.10)

where T− may not be equal to T+ , and want to know what happens in the limiting process. In fact, we find the solutions of compressible Navier–Stokes equations converge to a nonlinear diffusion wave solution globally in time as Mach number goes to zero. In particular, the velocity of diffusion wave is only driven by the variation of temperature. This phenomenon looks like thermal creep flow [16,18,28,41]. Moreover, when Mach number is small, the compressible Navier–Stokes system also has the same property, which has never been observed before.

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Now we begin to formulate the main results. The system (1.4) and the limiting system (1.7) in one dimension become, respectively, ⎧ ⎪ ∂t ρε + (ρε uε )x = 0, ⎪ ⎪ ⎨ ρε (uεt + uε uεx ) +

Pxε ε2

=μ ˜uεxx , ⎪ ⎪ ⎪ ⎩ρε (T ε + uε T ε ) + P ε uε = κT ε + μ ˜ε2 |uεx |2 , t x x xx

(1.11)

and ⎧ ⎪ (2u − κTx )x = 0, ρ = T −1 , ⎪ ⎪ ⎨ ρ(ut + uux ) + πx = μ ˜uxx , ⎪ ⎪ ⎪ ⎩ρ(T + uT ) + u = κT , t

x

x

(1.12)

xx

where (x, t) ∈ R × R+ and μ ˜ = 2μ + λ > 0. We shall construct a special solution of (1.12). Indeed, from (1.12)1 , we choose u=

κ κ ρx Tx = − 2 . 2 2ρ

(1.13)

Substituting (1.13) into (1.12), one obtains the following nonlinear diffusion equation ρt =

κ ρ  x

2 ρ

x

.

(1.14)

We consider the boundary condition at the far field, i.e., limx→±∞ ρ(x, t) = ρ± . From [4] and [12], it is known that the nonlinear diffusion equation (1.14) admits a unique selfx similar solution Ξ(η), η = √1+t satisfying Ξ(±∞, t) = ρ± . Let δ = |ρ+ − ρ− |, then Ξ(t, x) satisfies 2 O(1)δ − 4d(ρ±x)(1+t) κ . Ξx (t, x) =  , as x → ±∞, where d(ρ) = e 2ρ (1 + t)

We define . (¯ ρ, u ¯, T¯ ) =

κ Ξx −1 , Ξ, − , Ξ 2 Ξ2



(1.15)

which is a special solution of (1.12), that is ⎧ ⎪ (2¯ u − κT¯x )x = 0, ρ¯ = T¯ −1 , ⎪ ⎪ ⎨ ρ¯(¯ ut + u ¯u ¯x ) + π ¯x = μ ˜u ¯xx , ⎪ ⎪ ⎪ ⎩ρ¯(T¯ + u ¯T¯x ) + u ¯x = κT¯xx , t

(1.16)

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Fig. 1. 0 < T− < T+ .

Fig. 2. T− > T+ > 0.

with π ¯=μ ˜u ¯x − ρ¯u ¯2 + κ2 ρ¯ρ¯t . In particular, u ¯ = κ2 T¯x indicates that the flow is driven by the variation of temperature. In other word, the flow moves along the direction from the low temperature to the high one, see Figs. 1 and 2, and thus behaves like thermal creep flow. Indeed, Slemrod [41] mentioned: “This leaves open the issue of finding the correct extended fluid dynamics for moderate dense gas or polyatomic gas”, i.e., how to construct a thermal creep flow for moderate dense gas or polyatomic gas is still an open problem. In this paper, for well-prepared data, we will show that the solution of compressible Navier–Stokes equation behaviors as thermal creep flow in some sense when the Mach number is small. It should be noted that the usual Poiseuille flow moves from the high pressure part to the low one due to the difference of pressure. We will state our main results in the following two subsections. 1.1. Main results for well-prepared data In this subsection, we consider the low Mach limit around the diffusive wave with well-prepared initial data. It is more convenient to use the Lagrangian coordinates for the global behavior of solutions. That is, take the coordinate transformation (x,t)    (x, t) → ρε (z, s)dz − (ρε uε )(z, s)ds, s , (0,0)

which is still denoted as (x, t) without confusion. It is noted that the above transformation is independent of the path of integration due to the continuity equation (1.11)1 . We also point out that the Eulerian and Lagrangian formulations are equivalent since we consider the smooth solution of compressible Navier–Stokes equations, see also [5,40]. Let v = ρ1

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be the specific volume. Then, in Lagrangian coordinates, the system (1.11) can be written equivalently as ⎧ ⎪ v ε − uεx = 0, ⎪ ⎪ ⎨ t uε (1.17) ˜( vxε )x , uεt + ε12 Pxε = μ ⎪ ⎪ ⎪ ⎩(T ε + 1 |εuε |2 ) + (P ε uε ) = κ( 1 T ε ) + ε2 ( μ˜ uε uε ) , t x x x 2 vε x x vε where the pressure P = T /v. Similarly, the limiting system (1.16) in the Lagrangian coordinate becomes ⎧ ⎪ (2u − κv Tx )x = 0, v = T , ⎪ ⎪ ⎨ (1.18) u t + πx = μ ˜( v1 ux )x , ⎪ ⎪ ⎪ ⎩T + u = κ( 1 T ) . t

x

v x x

As in (1.15), we can construct a special diffusive wave solution (¯ v, u ¯, T¯ , π ¯ ) of (1.18) by choosing (¯ v, u ¯, T¯ ) := (Tˆ , where Tˆ (η), η = tion

√x 1+t

κTˆx ˆ , T ), 2Tˆ

(1.19)

is the unique self-similar solution of the following diffusion equa Tt =

κTx 2T

, with x

lim T = T± ,

x→±∞

(1.20)

and π ¯ can be solved by (1.18)2 . Let δ = |T+ − T− |, then Tˆ (t, x) satisfies x2 1 − κ , as x → ±∞. Tˆx (t, x) = O(1)δ(1 + t)− 2 e 4d(T± )(1+t) , d(T ) = 2T

(1.21)

Indeed, (¯ v, u ¯, T¯ ) is the Lagrangian version of diffusion wave solution (1.15). Since (¯ v, u ¯, T¯ ) is not the solution of (1.17) and some non-integrated error terms with respect to time t and slow ε-decay terms should appear in the low Mach limiting analysis, we need to introduce a new profile to approximate the system (1.17), i.e., 1 u|2 ). (˜ v, u ˜, T˜ ) := (¯ v, u ¯, T¯ − |ε¯ 2 Then a direct calculation gives that ⎧ ⎪ ˜x = 0, v˜t − u ⎪ ⎪ ⎨ ˜ 1x , ε2 u ˜t + P˜x = μ ˜ε2 ( u˜v˜x )x + R ⎪ ⎪

⎪ ˜ 1 ˜ ⎩ ˜ 2x , T + 2 |ε˜ u|2 t + (P˜ u ˜)x = (κ Tv˜x )x + ε2 ( μv˜˜ u ˜u ˜x )x + R

(1.22)

(1.23)

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where ˆ c± x2 ˜2 μ ˜ ˜ 1 = ε2 κTt − ε2 u − ε2 u ˜x = O(1)δε2 (1 + t)−1 e− 1+t , as |x| → ∞, R 2˜ v v˜ 2Tˆ 2 c± x2 3 ε 3 ε2 ˜ 2 = ε2 κ u ˜ − μ ˜u ˜u ˜x = O(1)δε2 (1 + t)− 2 e− 1+t , as |x| → ∞. R ˜u ˜x − u 2˜ v v˜ Tˆ

(1.24) (1.25)

It is noted that the system (1.23) approximate the original compressible Navier–Stokes ˜ 1x and R ˜ 2x , when ε is sufficiently small. equations (1.17) very well, up to error terms R Moreover, since we will integrate the differences between the solutions of original system (1.17) and the approximate system (1.23) with respect to space variable in (2.3) below, so it is very important that the error terms are in the form of x-derivatives. From (1.19) and (1.22), we also note that (¯ v − v˜, u ¯−u ˜, T¯ − T˜ )(t) ≤ Cε2 (1 + t)−1 ,

(1.26)

which implies that (˜ v, u ˜, T˜ ) approximate the diffusive wave solution (¯ v, u ¯, T¯ ) very well when ε is small. Now we supplement the system (1.17) with the initial data (v ε , uε , T ε )|t=0 = (˜ v, u ˜, T˜ )(x, 0),

(1.27)

then we have the following global existence and uniform estimates. Theorem 1.1 (Uniform estimates for well-prepared data). Let (˜ v, u ˜, T˜ )(x, t) be the diffusive wave defined in (1.22) with the wave strength δ = |T+ − T− |. There exist positive constants δ0 and ε0 , such that if δ ≤ δ0 and ε ≤ ε0 , then the Cauchy problem (1.17), (1.27) has a unique global smooth solution (v ε , uε , T ε ) satisfying ⎧ √ √ ⎨(v ε − v˜, εuε − ε˜ u, T ε − T˜ )(t)2L2 ≤ C δε3 (1 + t)−1+C0 δ , x √ √ 3 ⎩(v ε − v˜, εuε − ε˜ u, T ε − T˜ )x (t)2L2 ≤ C δε2 (1 + t)− 2 +C0 δ ,

(1.28)

x

where C and C0 are positive constants independent of ε and δ. Based on the uniform estimates (1.28) and Sobolev embedding, we justify the following low Mach limit. Corollary 1.2 (Low Mach limit for well-prepared data). Under the assumptions of Theorem 1.1, it follows from (1.26) and (1.28) that as ε → 0, ⎧ ⎨(v ε − v¯, T ε − T¯ )(t)L∞ (R) ≤ Cδ 14 ε 54 (1 + t)− 12 → 0, 1 1 1 ⎩(uε − u ¯)(t)L∞ (R) ≤ Cδ 4 ε 4 (1 + t)− 2 → 0.

(1.29)

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ˆ Note that the velocity u ¯ is driven by the variation of temperature T˜ , i.e., u ¯ = κ2TTˆx . Without loss of generality, we assume that T− < T+ , then for any given positive constant η0 > 0, there exists cη0 > 0 such that

Tˆ  (η) > cη0 δ > 0, for |η| ≤ η0 .

(1.30)

Moreover, we have Corollary 1.3 (Driven by the variation of temperature). For the solutions obtained in Theorem 1.1, there exists a small positive constant ε1 = ε1 (η0 ) ≤ ε0 , such that if ε ≤ ε1 , it follows from (1.29) and (1.30) that ⎧ ⎨0 < ⎩

cη0 δ √ C1 1+t

0<

<

1 ˆ 2 Tx

1 ˆ C1 Tx

≤

≤ uε (x, t) ≤ C1 Tˆx ,

Txε (x, t)

≤

3 ˆ 2 Tx ,

1

for |x| ≤ η0 (1 + t) 2 , t ≥ 0,

(1.31)

which yields immediately that 1 2 ε T (x, t) ≤ uε (x, t) ≤ 2C1 Txε (x, t), for |x| ≤ η0 (1 + t) 2 , t ≥ 0, 3C1 x

(1.32)

where C1 is a suitably large positive constant depending only on T± . Remark 1.4. The estimate (1.32) shows that the velocity uε of the compressible Navier– Stokes system (1.17) is also driven by the variation of temperature when ε is small. Note that the pressure P is almost 1, this phenomenon behaviors like thermal creep flow [16, 28,41]. So, our result can be regarded as an answer to the open question of [41] in some sense. Remark 1.5. All the above results for the well-prepared data hold globally in time. 1.2. Main results for ill-prepared data To understand the role of the thermodynamics, as in [2], we rewrite the equations (1.4) by the pressure fluctuations pε , and temperature fluctuations θε with ε

P ε (x, t) = eεp

(x,t)

, T ε (x, t) = eθ

ε

(x,t)

.

(1.33)

It follows from (1.2) and (1.33) that ε

ρε (x, t) = eεp

(x,t)−θ ε (x,t)

.

(1.34)

Under these changes of variables, the asymptotic states at infinity are, as x → ±∞ pε → 0, uε → 0, and θε → θ± , as x → ±∞, where θ± = ln T± .

(1.35)

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The compressible Navier–Stokes system (1.11) takes the following equivalent form ⎧ ε ε ε 1 ε ε ε ε −εpε +θ ε ε ⎪ θx )x = μ ˜εe−εp |uεx |2 + κe−εp +θ pεx · θxε , ⎪ ⎪pt + u · px + ε (2u − κe ⎨ e−θ [uεt + uε · uεx ] + 1ε pεx = μ ˜e−εp uεxx , ⎪ ⎪ ⎪ ε ⎩θε + uε θε + uε = κe−εpε (eθε θε ) + μ ˜ε2 e−εp |uεx |2 , t x x x x ε

ε

(1.36)

and the limiting system (1.12) becomes ⎧ ⎪ (2u − κeθ θx )x = 0, ⎪ ⎪ ⎨

e−θ (ut + uux ) + πx = (˜ μux )x , ⎪ ⎪ ⎪ ⎩θ + uθ + u = (κeθ θ ) . t

x

x

(1.37)

x x

Since θ− may not be equal to θ+ , we need to introduce a background profile θ˜ for θε . Here we choose θ˜ := − ln Ξ,

(1.38)

satisfying θ˜ → θ± as x → ±∞. Then we define the following solution space: for s ≥ 4, 2 ˜ N (t) := (pε , uε , θε − θ)(t) H s,ε

=

s 

∂ α (pε , uε )(t)2L2 +

|α|=0

+

s+1 

2 ˜ ∂ α (εpε , εuε )(t)2L2 + (θε − θ)(t) L2

|α|=0

∂ θ

α ε

(t)2L2

+

|α|=1

+

s+1 

s t 

∂ α (pεx , uεx )(τ )2L2 dτ

|α|=0 0

s+1 t 

∂ α (εuεx , θxε )(τ )2L2 dτ,

(1.39)

|α|=0 0

where ∂ α := (ε∂t )α0 ∂xα1 with multi-index α = (α0 , α1 ). We supplement the system (1.36) with the following initial data ε (pε , uε , θε )|t=0 = (pεin , uεin , θin ),

(1.40)

with 0 < a ≤ e−εpin ≤ a ¯, and 0 < b ≤ eθin ≤ ¯b, ε

ε

where a, a ¯, b and ¯b are given constants independent of ε. Then the uniform estimates for (1.36) are:

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Theorem 1.6 (Uniform estimate for ill-prepared data). Let s ≥ 4 be an integer, and ε assume that the initial data (pεin , uεin , θin ) satisfy ε ˜ 2 s,ε ≤ Cˆ0 < ∞, for ε ∈ (0, 1], (pεin , uεin , θin − θ) H

(1.41)

where Cˆ0 is independent of ε. Then there exist positive constants T0 and ε0 depending only on Cˆ0 and |θ+ − θ− | such that the Cauchy problem (1.36), (1.40) has a unique smooth solution (pε , uε , θε ) satisfying 2 ˜ ˜ (pε , uε , θε − θ)(t) H s,ε ≤ C0 , for t ∈ [0, T0 ], ε ∈ (0, ε0 ],

(1.42)

where the constant C˜0 > 0 depends only on Cˆ0 and |θ+ − θ− |. Remark 1.7. The time derivatives of the initial data in (1.41) are defined through the system (1.36). Remark 1.8. The wave strength |θ+ − θ− | (or equivalently |ρ+ − ρ− |) is allowed to be large. But the constants T0 and ε0 may depend on the wave strength. Then we have the following low Mach limit. Theorem 1.9 (Low Mach limit for ill-prepared data). Under the assumptions of Theorem 1.6 and further assume that the initial data satisfy ε ˜ → (pin , uin , θin − θ), ˜ in H s (R) as ε → 0, (pεin , uεin , θin − θ)

(1.43)

|θin (x) − θ+ | ≤ Cx−1−σ , for x ∈ [1, +∞),

(1.44)

where σ and C are some positive constants. Then the solutions (pε , uε , θε ) of (1.36) s (R)) for all s < s to (0, u, θ), where (u, θ) is the converge strongly in L2 (0, T0 ; Hloc unique solution of (1.37) with the initial data (win , θin ), where win is determined by win =

1 θin κe ∂x θin . 2

(1.45)

Remark 1.10. In the proof of Theorem 1.9, we extend the convergence lemma of [1,35] to the different asymptotic states at infinity. ε ˜u Remark 1.11. If θin − θ˜ → 0 as ε → 0, then one must have that θ¯ = θ, ¯ = which is indeed the diffusive wave solution constructed in (1.15).

1 θ˜ ˜ 2 κe ∂x θ

Remark 1.12. The condition (1.44) can also be replaced by |θin (x) − θ− | ≤ C|x|−1−σ , for x ∈ (−∞, −1].

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The paper is organized as follows. In Section 2 we consider the low Mach limit for the well-prepared data and in Section 3 for the ill-prepared data. Notations: Throughout this paper, the positive generic constants independent of ε are denoted by c, C, Ci (i ∈ N). And we will use  ·  to denote the standard L2 (R) norm, and  · H i (i ∈ N) to denote the Sobolev H i (R) norm. Sometimes, we also use O(1) to denote a uniform bounded constant independent of ε. 2. Well-prepared data This section is devoted to the low Mach limit for the well-prepared data. For simplicity, we omit the superscript ε of the variables throughout this section. We first reformulate the system (1.17) as follows. Reformulation of the system (1.17): Set the scaling y=

x , ε

τ=

t . ε2

(2.1)

In the following, we will also use the notations (v, u, T )(τ, y) and (˜ v, u ˜, T˜ )(τ, y), etc., in the scaled independent variables. Set the perturbation around the profile (˜ v, u ˜, T˜ )(τ, y) by   1 1 u|2 , T − T˜ (τ, y), (φ, ψ, ω, ζ)(τ, y) = v − v˜, εu − ε˜ u, T + |εu|2 − T˜ − |ε˜ 2 2

(2.2)

and introduce y ˜ )(τ, y) = (Φ, Ψ, W

(φ, ψ, ω)(τ, z)dz.

(2.3)

−∞

˜ )(0, ±∞) = 0. And it follows from the equations of From (1.27), we note that (Φ, Ψ, W ˜ ˜ )(τ, ±∞) ≡ 0 for τ > 0. (Φ, Ψ, W ) in (2.4) below that (Φ, Ψ, W Subtracting (1.23) from (1.17) and integrating the resulting system yield the following ˜ ): integrated system for (Φ, Ψ, W ⎧ ⎪ Φτ − Ψy = 0, ⎪ ⎪ ⎨

˜1, Ψτ + P − P˜ = μ ˜ v1 εuy − v1˜ ε˜ uy − R ⎪ ⎪



⎪ ⎩W ˜2. ˜ τ + εP u − εP˜ u ˜ = κ v1 Ty − v1˜ T˜y + μ ˜ε2 v1 uuy − v1˜ u ˜u ˜y − εR

(2.4)

˜ τ , while the dissipation We note that the term involving time derivative in (2.4)3 is W 1

1 ˜ term of (2.4)3 is in the form κ v Ty − v˜ Ty which is closely related to the perturbation

360

F. Huang et al. / Advances in Mathematics 319 (2017) 348–395

of temperature. To write the dissipation equation (2.4)3 in a unified way and make the linearized formulation more convenient to do the energy estimates, instead of the ˜ which is the anti-derivative of the total energy, it is more convenient to variable W introduce another variable W related to the temperature, i.e., ˜ − ε˜ W =W uΨ,

(2.5)

which yields that ζ = Wy − Y, with Y =

1 |Ψy |2 − ε˜ uy Ψ. 2

(2.6)

In terms of the new variable W , one linearize the system (2.4) as ⎧ ⎪ Φτ − Ψy = 0, ⎪ ⎪ ⎨

Ψτ − v1˜ Φy + v1˜ Wy = μ ˜ v1˜ Ψyy + Q1 , ⎪ ⎪ ⎪ ⎩W + Ψ = κ W + Q , τ y yy 2 v ˜

(2.7)

where 1 1 1 ˜1, − εuy + J1 + Y − R v v˜ v˜ 1 1 εuy κ ˜ 2 + ε˜ ˜1, Ψy + J2 − ε˜ Q2 = κ − Ty + μ ˜ uτ Ψ − Yy − εR uR v v˜ v v˜

Q1 = μ ˜

(2.8) (2.9)

and ⎧ ˜ ˜ P˜ (v − v˜) − 1 (T − T˜ )] ⎪ J1 = P v−1 ⎪ ˜ Φy − [P − P + v ˜ v ˜ ⎪ ⎨ 2 2 2 4 = O(1)(|Φy | + |Wy | + Y + |ε˜ u| ), ⎪ ⎪ ⎪ ⎩J = (1 − P )Ψ = O(1)(|(Φ , Ψ , W )|2 + Y 2 + |ε˜ u|4 ). 2

y

y

y

y

To prove Theorem 1.1, one needs only to show the following a priori estimate in the scaled independent variables and employ the continuity argument since the local existence for compressible Navier–Stokes system is known. Proposition 2.1 (A priori estimates). Assume that (Φ, Ψ, W ) is a smooth solution of (2.7) with zero initial data in the time interval τ ∈ [0, T ]. There exist constants δ1 > 0 and ε1 > 0 such that if δ ≤ δ1 and ε ≤ ε1 , the following estimates hold ⎧ √ ⎪ ≤ C δε, (Φ, Ψ, W )(τ )2L∞ ⎪ y ⎪ ⎨ √ √ (φ, ψ, ζ)(τ )2L2 ≤ C δε2 (1 + ε2 τ )−1+C0 δ , y ⎪ ⎪ √ √ ⎪ 3 ⎩ (φy , ψy , ζy )(τ )2L2 ≤ C δε3 (1 + ε2 τ )− 2 +C0 δ . y

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To obtain the above a priori estimates, one assumes the following a priori assumption: N (T ) = sup

1

0≤τ ≤T

ε

(Φ, Ψ, W )(τ )2L∞ +

 1 1 2 2 (φ, ψ, ζ)(τ ) (φ , ψ , ζ )(τ ) 2 + 2 y y y L L ε2 ε3

≤χ , 2

(2.10)

where χ is a small constant determined later. It is noted that the local existence theorem yields that (2.10) holds for some positive time which may depend on ε > 0 and δ > 0. We also point out that once we have proved the above Proposition 2.1, then it implies that (2.10) is valid, and hence we justify the use of above a priori assumption (2.10). Basic estimates: We start with the elementary energy estimates. Multiplying (2.7)1 by Φ, (2.7)2 by v˜Ψ, (2.7)3 by W respectively, and adding all the resulting equations, one can obtain that

1 2 1 2 1 2 Φ + v˜Ψ + W 2 2 2

+μ ˜Ψ2y + τ

κ 2 W v˜ y

κ 1 = v˜τ Ψ2 + v˜Q1 Ψ + Q2 W − Wy W + (· · · )y . 2 v˜ y

(2.11)

Define m = (Φ, Ψ, W )t , where (·, ·, ·)t is the transpose of the vector (·, ·, ·), then from (2.7), one has mτ + A1 my = A2 myy + A3 ,

(2.12)

where ⎛

0

⎜ A1 = ⎝ − v1˜ 0

−1

0

0

1 v ˜

1

0

⎞ ⎟ ⎠,

⎛

0

⎜ A2 = ⎝ 0 0

0

0

⎞

μ ˜ v ˜

⎟ 0 ⎠ , A3 = (0, Q1 , Q2 )t .

0

κ v ˜

It follows from a direct computation that the eigenvalues of the matrix A1 are λ1 , 0, λ3  2 with λ3 = −λ1 = v˜ . The corresponding normalized left and right eigenvectors can be chosen as  2 2 1 1 1 (1, 0, 1), l3 = (−1, , 1), l1 = (−1, − , 1), l2 = 2 λ3 2 2 λ3  1 1 1 t (1, 0, 1)t , r3 = (−1, λ3 , 1)t r1 = (−1, −λ3 , 1) , r2 = 2 2 2 so that

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⎛

li rj = δij , i, j = 1, 2, 3,

λ1 ⎜ LA1 R = Λ = ⎝ 0 0

0 0 0

⎞ 0 ⎟ 0 ⎠, λ3

where L = (l1 , l2 , l3 )t , R = (r1 , r2 , r3 ). Let B = Lm = (b1 , b2 , b3 ),

(2.13)

then multiplying the equations (2.12) by the matrix L yields that Bτ + ΛBy = LA2 RByy + 2LA2 Ry By + [(Lτ + ΛLy )R + LA2 Ryy ]B + LA3 .

(2.14)

We shall use a weighted energy method to derive the intrinsic dissipation. Without loss of generality, we assume that Tˆy > 0. The case that Tˆy < 0 can be treated in the same ˆ ˜ = (T N b1 , b2 , T −N b3 ) way. Let T1 = TT+ , then |T1 − 1| ≤ Cδ. Multiplying (2.14) by B 1 1 with a large positive integer N which will be chosen later, one can get that     N

T1N 2 1 2 T1−N 2 T1 T1−N 2 ˜y A4 By + BA ˜ 4y By b + b + b − b − b23 + B 2 1 2 2 2 3 2 τ 1 2 τ

−

T1N −1

τ

T1−N −1

(N λ1 T1y + T1 λ1y )b21 + (N λ3 T1y − T1 λ3y )b23 + (· · · )y 2 2 ˜ ˜ ˜ ˜ = 2BLA 2 Ry By + B[Lτ R + LA2 Ryy ]B + BΛLy RB + BLA3 .

(2.15)

A direct computation shows that the symmetric matrix LA2 R = A4 is nonnegative. Define     N 1 2 1 2 1 2 T1 2 1 2 T1−N 2  Φ + v˜Ψ + W dy + b + b + b dy, E1 = (2.16) 2 2 2 2 1 2 2 2 3    κ μ ˜Ψ2y + Wy2 + By A4 By dy. (2.17) K1 = v˜ Note that

      ˜ − B)y A4 By dy  ≤ Cδ |By |2 dy + Cδ |Tˆy |2 |B|2 dy  (B    ≤ Cε2 δ(1 + t)−1 E1 + CδK1 + Cδ |Φy |2 dy.

(2.18)

˜ 4y By , BLA ˜ ˜ Similarly, the terms BA 2 Ry By and B[Lτ R + LA2 Ryy ]B in (2.15) satisfy the ˜ ˜ same estimate. For BΛLy RB and BLA3 , we need to use the explicit presentation. By the choice of the characteristic matrix L and R, one has that

F. Huang et al. / Advances in Mathematics 319 (2017) 348–395

⎛

1 0 1 ⎜ ΛLy R = λ3y ⎝ 0 0 2 1 0

363

⎞  t  −1 1 Q1 1 1 Q1 ⎟ Q2 − Q2 , Q2 + , , 0 ⎠ , and LA3 = 2 λ3 2 2 λ3 −1

which yields immediately that ˜ y RB| ≤ C|λ3y |(b21 + b23 ), |BΛL

˜ BLA 3 ≤ C|(b1 , b3 )| · |(Q1 , Q2 )| + C|b2 Q2 |. (2.19)

Integrating (2.11) over R with respect to y, using (2.15)–(2.18), (2.19) and the Cauchy inequality, and choosing N sufficiently large, we obtain that 1 E1τ + K1 +2 2



|Tˆy |(b21 +b23 )|dy ≤ Cε2 δ(1+t)−1 (E1 +1)+CδK1 +Cδ

 Φ2y dy+I, (2.20)

with  I=C

|(b1 , b3 )| · |(Q1 , Q2 )| + |b2 Q2 |dy.

(2.21)

Here, in the proof of (2.20), we have used the fact Ψ=

1 λ3 (b3 − b1 ), 2

(2.22)

and for N large enough that 1 1 ˜ y RB − T1N −1 (N λ1 Tˆ1y + 2T1 λ1y )b21 + T1−N −1 (N λ3 T1y − 2T1 λ3y )b23 − BΛL 2 2 (2.23) ≥ 3|Tˆy |(b2 + b2 ). 1

3

˜ 1 with the decay rate ε2 (1 + t)−1 , the terms in (2.21) Although Q1 contains the term R involving Q1 can be estimated by the intrinsic dissipation on b1 and b3 as shown later. ˜ 2 which has a better decay rate ε3 (1 + t)− 32 The terms involving Q2 contains the term εR   and can be estimated directly. For brevity, we only estimate |Q1 b1 |dy and |Q2 b2 |dy as follows for illustration.  Estimation on |Q1 b1 |dy: It follows from (2.8) that  |Q1 b1 |dy ≤

   ˜ μ ˜ Y  μ  ˜ 1 b1 |dy =: I1 + I2 . − uy + J1 + · b1  + |R  v v˜ v˜

(2.24)

A direct calculation shows that    μ  ˜  ˜−μ εuy · b1 dy ≤ C(χ + δ)(K1 + Φy 2L2 + ψy 2L2 ) + Cδε4 (1 + t)−2 Φ2L2 v v˜ + Cδε2 (1 + t)−1 E1 + Cδε5 (1 + t)− 2 , 5

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and  

|J1 | + |

Y  | · |b1 |dy ≤ C(δ + χ)(K1 + Φy 2L2 + ψy 2L2 ) + Cδε4 (1 + t)−2 Φ2L2 v˜ + Cδε2 (1 + t)−1 E1 + Cδε5 (1 + t)− 2 , 5

which yields immediately that I1 ≤ C(δ + χ)(K1 + Φy 2L2 + ψy 2L2 ) + +Cδε2 (1 + t)−1 E1 + Cδε5 (1 + t)− 2 . (2.25) 5

On the other hand, it follows from (1.24) and the Cauchy inequality that 

 ˜ 1 b1 |dy ≤ δ |R

|Tˆy |b21 dy + Cδε2 (1 + t)−1 ,

which, together with (2.25), yields that   |Q1 b1 |dy ≤ δ |Tˆy |b21 dy + C(δ + χ)(K1 + Φy 2L2 + ψy 2L2 ) + Cδε2 (1 + t)−1 (E1 + 1).

(2.26)

 Estimation on |Q2 b2 |dy: It follows from (2.9) that  |Q2 b2 |dy ≤

   κ κ  μ ˜ κ ˜ 2 +ε˜ ˜ 1  ·|b2 |dy. (2.27)  − uτ Ψ− Yy −εR uR Ty + εuy Ψy +J2 −ε˜ v v˜ v v˜

The Cauchy inequality yields that    κ κ  μ ˜  − Ty + εuy Ψy  · |b2 |dy v v˜ v ≤ C(δ + χ)(K1 + Φy 2L2 ) + Cχ(ψy , ζy )2L2 + Cδε2 (1 + t)−1 E1 ,

(2.28)

and 

  κ ˜ 2 + ε˜ ˜ 1  · |b2 |dy J2 − ε˜ uτ Ψ − Yy − εR uR v˜

≤ C(δ + χ)(K1 + Φy 2L2 ) + Cχψy 2L2 + Cδε2 (1 + t)−1 (E1 + 1).

(2.29)

Thus combining (2.28), (2.29) and using the Cauchy inequality, one has that  |Q2 b2 |dy ≤ C(δ + χ)(K1 + Φy 2L2 ) + Cχ(ψy , ζy )2L2 + Cδε2 (1 + t)−1 (E1 + 1).

(2.30)

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Substituting (2.26), (2.30) into (2.20), one obtains that E1τ

1 + K1 + 4

 |Tˆy |(b21 + b23 )|dy

≤ Cδε2 (1 + t)−1 (E1 + 1) + C(δ + χ)(Φy 2L2 + (ψy , ζy )2L2 ).

(2.31)

Note that the norm Φy 2L2 is not included in K1 . To complete the lower-order inequality, we need to estimate Φy . Using (2.7)1 , we rewrite (2.7)2 to be the following form 1 1 μ ˜ Φyτ − Ψτ + Φy = Wy − Q1 . v˜ v˜ v˜

(2.32)

Multiplying (2.32) by Φy yields that μ 1 μ  ˜ 2 1 ˜ 2 Φy − Wy − Q1 Φy . Φy − Φy Ψτ + Φ2y = 2˜ v 2˜ v τ v˜ v˜ τ

(2.33)

Noting Ψτ Φy = (ΨΦy )τ − (ΨΦτ )y + Ψ2y , which, together with (2.33) and integrating by parts, yields that     1  μ ˜ 2 2 2 2 −3/2 Φ −Φy Ψdy + Φ dy ≤ C Ψy +Wy dy +Cδ(1+t) + Q21 dy. (2.34) 2˜ v y 2˜ v y τ On the other hand, it follows from (2.8) and (2.10) that 

Q21 dy ≤ C(δ + χ)(K1 + Φy 2 + ψy 2 ) + Cδε2 (1 + t)−1 E1 + Cδε2 (1 + t)−1 . (2.35)

Substituting (2.35) into (2.34), one has that    μ 1 2 ˜ 2 Φy − Φy Ψdy + Φ dy 2˜ v 4˜ v y τ ≤ C1 K1 + C1 δε2 (1 + t)−1 (E1 + 1) + C1 (δ + χ)ψy 2 .

(2.36)

By choosing C˜1 large enough, it holds that  C˜1 E1 +

μ ˜ 2 1 Φy − Φy Ψdy ≥ C˜1 E1 + 2˜ v 2



μ ˜ 2 1 1 Φy dy, and C˜1 − C1 ≥ C˜1 . 4˜ v 4 8

(2.37)

Thus, multiplying (2.31) by C˜1 and using (2.37), one obtains the low order estimates:

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Lemma 2.2. If δ and χ are suitably small, then it holds that 

|Tˆy |(b21 + b23 )|dy ≤ Cδε2 (1 + t)−1 (E1 + 1) + C(δ + χ)(ψy , ζy )2 ,

E2τ + K2 + where

 E2 = C˜1 E1 +

μ ˜ 2 1 Φy − Φy Ψdy, K2 = C˜1 K1 + 2˜ v 8



1 2 Φ dy. 8˜ v y

Derivative estimates: To obtain the estimate for the first-order derivative of (Φy , Ψy , Wy ), we shall use an energy estimate based on the convex entropy of the compressible Navier–Stokes equations. Applying ∂y to (2.4) yields that ⎧ ⎪ φτ − ψy = 0, ⎪ ⎪ ⎨ ˜ 1y , ˜y )y − R ψτ + (P − P˜ )y = ε( μv˜ uy − μv˜˜ u ⎪ ⎪ ⎪ ⎩ζ + ε(P u − P˜ u ˜y ) = ( κv Ty − κv˜ T˜y )y + Q3 , τ y

(2.38)

where Q3 =

μ  μ ˜ 2 2 1 ˜ ˜ 2y . ε uy − ε2 u ˜u ˜y + εP˜y u ˜ + (ε2 u ˜2 )τ − εR v v˜ 2 y

Lemma 2.3. If δ and χ are suitably small, then it holds that E3τ

1 + K3 ≤ Cδε2 (1 + t)−1 E3 + ε2 (1 + t)−1 2



|Tˆy |(b21 + b23 )dy + Cδε4 (1 + t)−2 , (2.39)

where  E3 =

1 2 ˜ ˆv  ˜ ˆT  ψ +TΦ dy, +TΦ 2 v˜ T˜

 K3 =

μ ˜ 2 3 κ 2 ψ + ζ dy. v y 4 vT y

Proof. Multiplying (2.38)2 by ψ yields that 1 2

ψ2

μ ˜

 τ

− (P − P˜ )ψy + ε

v

uy −

˜  μ(θ) ˜ 1y ψ + (· · · )y . u ˜y ψy = −R v˜

Since P − P˜ = T˜ ( v1 − v1˜ ) + vζ , as in [17], it holds that 1 2

ψ2

 τ

− T˜

1 v

−

μ 1 ˜ ζ μ ˜ ˜ μ ˜ 1y ψ + (· · · )y . φτ − ψy + ψy2 + − ε˜ u y ψ y = −R v˜ v v v v˜

(2.40)

Denote ˆ Φ(s) = s − 1 − ln s,

(2.41)

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ˆ  (1) = 0 and Φ(s) ˆ is strictly convex around then, it is straightforward to check that Φ s = 1. Moreover, it holds that 

          ˆ v + T˜ − 1 + 1 φτ + T˜ − v + 1 v˜τ + T˜ − 1 + 1 v˜τ ˆ v = T˜τ Φ T˜ Φ 2 v˜ τ v˜ v v˜ v˜ v˜ v v˜ v  1 1 v  ˆ ˆ = T˜ − + φτ − P˜ Ψ v˜τ + v˜P˜τ Φ , (2.42) v v˜ v˜ v˜

ˆ ˆ −1 ). Therefore, it follows from (2.40) and (2.42) that with Ψ(s) = Φ(s     ˜ ˜ 2 μ ˜ μ ˆ v − ζ ψy + μ ˆ v ψ 2 + T˜ Φ ψy + − + P˜ v˜τ Ψ ε˜ u y ψy 2 v˜ τ v˜ v v v v˜ v  ˆ ˜ 1y ψ + (· · · )y . = v˜P˜τ Φ −R v˜ 1

On the other hand, multiplying (2.38)3 by

ζ T

(2.43)

yields that

κ ζ κ  ζ ζ ζ ζτ + ε(P uy − P˜ u Ty − T˜y + Q3 . ˜y ) = T T v v˜ T yT

(2.44)

A direct calculation gives that   ζ ˆ T ) + O(1)|T˜τ |ζ 2 , ζτ = T˜ Φ( T T˜ τ ζ ζ ζ ε(P uy − P˜ u ˜y ) = ψy + (P − P˜ )ε˜ uy , T v T

(2.45) (2.46)

and κ v

Ty −

3 κ 2 κ˜ ζ Ty ≤− ζ + O(1)|T˜y |2 |(φ, ζ)|2 + (· · · )y . v˜ 4 vT y yT

(2.47)

Substitute (2.45)–(2.47) into (2.44), one obtains that 

  ζ 3 κ 2 ζ ˆ T ζ ≤ Cδε2 (1 + t)−1 |(φ, ζ)|2 + Q3 + (· · · )y . + ψy + T˜ Φ 4 vT y T T˜ τ v

(2.48)

It follows from (2.48) and (2.43) that 1

v

 T 

μ ˜ 2 3 κ 2 ψ + ζ v y 4 vT y ˜ 1y ψ + Q3 ζ + (· · · )y . ≤ Cδε2 (1 + t)−1 |(φ, ζ)|2 − R T 2

ˆ ψ 2 + T˜ Φ

v˜

ˆ + T˜ Φ

T˜

τ

+

(2.49)

Using (2.22), one has that          2 −1 ˜ ˜ |Tˆy |(b21 + b23 )dy + Cδε4 (1 + t)−2 , (2.50)  R1y ψdy  =  R1yy Ψdy  ≤ ε (1 + t)

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and

   cx2 ζ   2 4 Q dy ≤ C |ζ| · |ψ | dy + Cδε |ζ|(1 + t)−2 e− 1+t dy   3 y θ ≤ Cχψy 2 + Cδε2 (1 + t)−1 ζ2 + Cδε5 (1 + t)− 2 . 5

(2.51)

Integrating (2.49) with respect to y, using (2.50) and (2.51), then one obtains (2.39). Thus, the proof of Lemma 2.3 is completed. 2 Since the norm φy 2 is not included in K3 . To complete the first derivative inequality, we follow the same way for Φy . We rewrite the equation (2.38)2 as μ  μ  ˜ ˜ ˜ μ μ ˜ ˜ 1y . φyτ − ψτ + (P − P˜ )y = − − φy − εuy + R v˜ v˜ y v v˜ y

(2.52)

Lemma 2.4. If δ and χ are suitably small, then it holds that  ˜   μ P 2 ˜ 2 φy − φy ψdy + φ dy 2˜ v 2˜ v y τ ≤ C2 K3 + C2 δε2 (1 + t)−1 E3 + C2 δε5 (1 + t)− 2 + C2 ((φ, ψ)2L∞ + ψy 2 )ψyy 2 . (2.53) 5

Proof. Multiplying (2.52) by φy yields that μ μ  ˜ ˜ 2  ˜ ˜ μ ˜ 2 μ μ ˜ 1y φy . φy − ψτ φy − (P − P˜ )y φy = φy + − ( )y ψy − [( − )εuy ]y φy + R 2˜ v 2˜ v τ v˜ v v˜ τ A direct calculation yields that P 1 1 P˜  P˜ 1 − − −(P − P˜ )y = φy − ζy + vy − Ty , v˜ v˜ v v˜ v v˜ and φy ψτ = (φy ψ)τ − (φτ ψ)y + ψy2 . Then combining the above three equations, one obtains that  ˜   μ P 2 ˜ 2 φy − φy ψdy + φ dy 2˜ v 2˜ v y τ ≤ C2 K3 + C2 δε2 (1 + t)−1 E3 + C2 δε5 (1 + t)− 2 + C2 ((φ, ψ)2L∞ + ψy 2 )ψyy 2 , (2.54) 5

where we have used  −

 (P − P˜ )y φy dy ≥

3P˜ 2 φ dy − Cζy 2 − Cδε2 (1 + t)−1 E3 , 4˜ v y

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369

 μ   μ    ˜ ˜ ˜ μ      ˜ 1y φy dy   ( )y ψy dy  +  [( − )εuy ]y φy dy  +  R v˜ v v˜  ˜ 5 P 2 ≤ φy dy + CK3 + Cδε2 (1 + t)−1 E3 + C2 δε5 (1 + t)− 2 8˜ v  + |φy |2 |ψy | + |φy ψy ζy |dy, and 

1

1

1

1

|φy |2 |ψy | + |φy ψy ζy |dy ≤ Cφy 2 ψy  2 ψyy  2 + Cφy ζy ψy  2 ψyy  2  ≤

P˜ 2 φ dy + Cζy 2 + Cψy 2 ψyy 2 . 16˜ v y

Therefore the proof of Lemma 2.4 is completed. 2 We now derive the higher order estimates. Applying ∂y to (2.38) yields that ⎧ ⎪ ⎪φyτ − ψyy = 0, ⎪ ⎨ ˜ 1yy , ψyτ + v1˜ ζyy − v1˜ φyy = μ ˜( v1 εuy − v1˜ ε˜ uy )yy + Q4 − R ⎪ ⎪ ⎪ ⎩ζ + ψ = ( κ T − κ T˜ ) + Q , yτ

yy

v

y

v ˜ y yy

(2.55)

5

where Q4 =

P 1  1 P˜  P˜ − 1 1 1  φyy + − φyy − 2 2 Ty vy − 2 T˜y v˜y + 2 3 vy2 − 3 v˜y2 v˜ v v˜ v v˜ v v˜   + O(1) |˜ vyy ||(φ, ψ)| + |φζyy | , (2.56)

Q5 = −ε˜ uyy (P − P˜ ) − ε(Py uy − P˜y u ˜y ) + (1 − P˜ )ψyy + Q3y .

(2.57)

Lemma 2.5. If δ and χ are suitably small, then it holds that 1 3 10 1 E4τ + K4 ≤ Cδε(1 + t)− 2 (φy , ψy , ζy )2 + Cδε3 (1 + t)− 2 E3 + C(φy , ψy , ζy ) 3 2

+ C(φ, ψ) · (φy , ψy , ζy )3 + Cδε6 (1 + t)−3 ,

(2.58)

where  E4 =

1 2 1 2 1 2 φ + v˜ψ + ζ dy, K4 = 2 y 2 y 2 y



μ ˜v˜ 2 κ 2 ψyy + ζyy dy. v v

Proof. Multiplying (2.55)1 by φy , (2.55)2 by v˜ψy , (2.55)3 by ζy and adding the resulting equations, one obtains that

370

F. Huang et al. / Advances in Mathematics 319 (2017) 348–395



1 2 1 2 1 2 φ + v˜ψ + ζ 2 y 2 y 2 y

+ τ

μ ˜v˜ 2 κ ψ + ζ2 v yy v yy

1 ˜ 1yy v˜ψy + Q5 ζy + J3 + J4 + (· · · )y , = v˜τ ψy2 + v˜Q4 ψy − R 2

(2.59)

where 1

μ  μ   ˜ 1 ˜ ˜ μ εuy − ε˜ uy v˜y ψy − − ψyy ψy − ε˜ uy v˜ψyy , v v˜ v y v v˜ y y κ  κ κ   − J4 = − ζyy ζy − θ˜y ζyy . v y v v˜ y

J3 = −˜ μ

It is direct to obtain    P˜ − 1  P  P˜    φyy + − φyy v˜ψy dy   v˜ v v˜    P P˜    − = (P˜ − 1)ψy + v˜ψy φy dy  v v˜ y ≤ (δ + χ)ψyy 2 + Cδε3 (1 + t)− 2 (φy , ψy , ζy )2 + Cδε4 (1 + t)−2 E3 3

10

+ C(φy , ψy , ζy ) 3 + C(φ, ψ)(φy , ψy , ζy )3 ,   1    1  1 ˜  1 2     2 T 2 T v − v ˜ dy + 2 v − v ˜ dy v ˜ ψ v ˜ ψ    y y y y y y y y v2 v˜2 v3 v˜3

(2.60)

≤ χψyy 2 + Cδε(1 + t)− 2 (φy , ψy , ζy )2 + Cδε3 (1 + t)− 2 E3 1

3

10

+ C(φy , ψy , ζy ) 3 ,

(2.61)

and 



v ψy |dy |˜ vyy ||(φ, ψ)| + |φζyy | |˜

≤ χζyy 2 + Cδε(1 + t)− 2 (φy , ψy , ζy )2 1

+ Cδε3 (1 + t)− 2 E3 + C(φ, ψ)(φy , ψy , ζy )3 . 3

(2.62)

Then it follows from (2.56) and (2.60)–(2.62) that   1 3    v˜Q4 ψy dy  ≤ (δ + χ)(ψyy , ζyy )2 + Cδε(1 + t)− 2 (φy , ψy , ζy )2 + Cδε3 (1 + t)− 2 E3 10

+ C(φ, ψ)(φy , ψy , ζy )3 + C(φy , ψy , ζy ) 3 .

(2.63)

Using Cauchy inequality, it is easy to get that    ˜ 1yy ψy dy  ≤ Cδε(1 + t)− 12 ψy 2 + Cδε6 (1 + t)−3 .  v˜R

(2.64)

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371

A direct calculation also yields that   1    Q5 ζy dy  ≤ (δ + χ)(ψyy , ζyy )2 + Cδε(1 + t)− 2 (φy , ψy , ζy )2 + Cδε3 (1 + t)− 2 E3 + C(φy , ψy , ζy ) 3 , 3

and



10

(2.65)

|J3 | + |J4 |dy ≤ (δ + χ)(ψyy , ζyy )2 + Cδε(1 + t)− 2 (φy , ψy , ζy )2 1

+ Cδε3 (1 + t)− 2 E3 + C(φy , ψy , ζy ) 3 . 3

10

(2.66)

Integrating (2.59) with respect to y and using (2.63)–(2.66), one obtains (2.58). Therefore the proof of Lemma 2.5 is completed. 2 We choose a large constants C˜2 > 1 so that   ˜ ˜ 2μ(θ) 1˜ 2μ(θ) 1˜ 1 2 ˜ φy − ψφy dy ≥ C2 E3 + φ2y dy, C2 − C2 ≥ C˜2 , C2 E3 + 3˜ v 2 3˜ v 2 4 and define E5 = C˜2 E3 +



˜ 2μ(θ) 1 φ2 − ψφy dy + ε−1 E4 , K5 = C˜2 K3 + 3˜ v y 4



(2.67)

p˜ 2 1 φ dy + ε−1 K4 . 4˜ v y 2 (2.68)

Thus, combining (2.39), (2.53) and (2.58), one obtains Lemma 2.6. If δ and χ are suitably small, then it holds that  3 E5τ + K5 ≤ C3 ε2 (1 + t)−1 |Tˆy |(b21 + b22 )dy + C3 δε2 (1 + t)− 2 E5 + C3 δε4 (1 + t)−2 . (2.69) Proof of Proposition 2.1. Let C˜3 = 1 + C3 , then from (2.36) and (2.69), one gets that √ √ E6τ + K6 ≤ C0 δε2 (1 + ε2 τ )−1 (E6 + δ), where 1 1 E6 = C˜3 E2 + ε−2 E5 , K6 = C˜3 K2 + ε−2 K5 . 4 2 Then the Gronwall’s inequality yields that √

E6 (τ ) ≤ C0 δ(1 + ε τ ) 2

C0

√

τ δ

,

√ √ K6 (s)ds ≤ CC0 δ(1 + ε2 τ )C0 δ .

0

Since c3 (Φ, Ψ, W )2 ≤ E6 for some positive constant c3 , one obtains that

(2.70)

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372

√ √ (Φ, Ψ, W )2 ≤ CC0 δ(1 + ε2 τ )C0 δ .

(2.71)

Multiplying (2.69) by (1 + ε2 τ ), one has that  [(1 + ε2 τ )E5 ]τ + (1 + ε2 τ )K5 ≤ Cε2

|Tˆy |(b21 + b22 )dy + Cε2 E5 + C3 δε4 (1 + ε2 τ )−1

≤ Cε2 K6 + C3 δε4 (1 + ε2 τ )−1 ,

(2.72)

where we have used the fact that E5 (τ ) ≤ C(φ, ψ, ζ)2 + Cε−1 (φy , ψy , ζy )2 ≤ C(Φy , Ψy , Wy )2 + Cε−1 (φy , ψy , ζy )2 + Cδε4 (1 + ε2 τ )− 2 3

≤ CK6 + Cδε4 (1 + ε2 τ )− 2 . 3

Integrating (2.72) over [0, τ ] and using (2.70), one obtains that √

−1+C0

E5 ≤ CC0 δε (1 + ε τ ) 2

2

√

τ δ

,

√ √ (1 + ε2 s)K5 ds ≤ CC0 δε2 (1 + ε2 τ )C0 δ . (2.73)

0

Furthermore, it holds that E5 (τ ) ≥ c4 (φ, ψ, ζ)2 + c4 ε−1 (φy , ψy , ζy )2 ≥ c4 (Φy , Ψy , Wy )2 + c4 ε−1 (φy , ψy , ζy )2 − c4 δε4 (1 + ε2 τ )− 2 , 3

(2.74)

for some small positive constant c4 . Then, it follows from (2.73) and (2.74) that ⎧ √ √ ⎨(Φy , Ψy , Wy )2 + (φ, ψ, ζ)2 ≤ CC0 δε2 (1 + ε2 τ )−1+C0 δ , √ ⎩(φ , ψ , ζ )2 ≤ CC √δε3 (1 + ε2 τ )−1+C0 δ . y y y 0

(2.75)

From (2.71) and (2.75), one has the decay rate for (Φ, Ψ, W ), (Φ, Ψ, W )L∞ ≤ C(Φ, Ψ, W ) 2 (Φy , Ψy , Wy ) 2 ≤ CC0 δ 4 ε 2 (1 + ε2 τ )− 4 + 2 C0 1

1

1

1

1

1

√

δ

. (2.76)

However, from (2.58), one observes that (φy , ψy , ζy )2 should have better time-decay 3 rate than the one in (2.75). In fact, multiplying (2.58) by (1 + ε2 τ ) 2 , one gets that 3

[(1 + ε2 τ ) 2 E4 ]τ ≤ Cδε3 (Φy , Ψy , Wy )2 + Cε(1 + ε2 τ )(φy , ψy , ζy )2 + Cδε6 (1 + ε2 τ )− 2 + C(1 + ε2 τ ) 2 (φy , ψy , ζy ) 3 3

3

3

+ C(1 + ε2 τ ) 2 (φ, ψ)(φy , ψy , ζy )3 .

10

(2.77)

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373

By using (2.73) and (2.75), one has that τ

3 2

τ

10 3

(1 + ε s) (φy , ψy , ζy ) ds + 2

0

3

(1 + ε2 s) 2 (φ, ψ)(φy , ψy , ζy )3 ds 0

τ ≤ ε2

(1 + ε2 s) 2 [(1 + ε2 s)− 3 + 3 C0 3

2

2

√

δ

+ (1 + ε2 s)−1+C0

√

δ

](φy , ψy , ζy )2 ds

0

τ ≤ ε2

√ √ (1 + ε2 s)(φy , ψy , ζy )2 ds ≤ CC0 δε4 (1 + ε2 τ )C0 δ ,

(2.78)

0

√ for C0 δ ≤ 18 . Thus, integrating (2.77) over [0, τ ] and using (2.78), (2.73) and (2.75), one gets that √ √ 3 (φy , ψy , ζy )2 ≤ CE4 ≤ CC0 δε3 (1 + ε2 τ )− 2 +C0 δ .

By using Sobolev inequality, we obtain 1

1

(φ, ψ, ζ)L∞ ≤ (φ, ψ, ζ) 2 (φy , ψy , ζy ) 2 ≤ Cδ 4 ε 4 (1 + ε2 τ )− 8 + 2 C0 1

5

5

1

√

δ

≤ Cδ 4 ε 4 (1 + ε2 τ )− 16 , 1

5

9

√ for C0 δ ≤ 18 . Therefore the a priori assumption (2.10) is closed and the proof of Proposition 2.1 is completed. 2 3. Ill-prepared data 3.1. Uniform estimates For simplicity, throughout this subsection we omit the superscript ε of the variables and denote a(εp) = e−εp , b(θ) = eθ .

(3.1)

For later use, we define that  ⎧ ⎪ Q(t) := sup 0≤τ ≤t (p, u)(τ )Hs + (εp, εu)(τ )Hs+1 ⎪ ⎪ ⎨  ˜ )L2 + ((ε∂t )θ, θx )(τ )Hs , + (θ − θ)(τ ⎪ ⎪ ⎪ ⎩ S(t) := (px , ux )(t)Hs + (εux , θx )(t)Hs+1 , where we have used the notation f (t)Hs := (ε∂t )α0 ∂xα1 and α = (α0 , α1 ).

 |α|≤s

(3.2)

∂ α f (t)L2 (R) with ∂ α :=

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374

We assume the following a priori assumption 0<

1 1 a ≤ a(εp) ≤ 2¯ a, 0 < b ≤ b(θ) ≤ 2¯b, on t ∈ [0, T ]. 2 2

(3.3)

To prove Theorem 1.6, we need only to prove the following a priori estimates. Proposition 3.1. For any given integer s ≥ 4 and ε ∈ (0, 1], let (pε , uε , θε ) be the classical solution to the Cauchy problem (1.36) and (1.40). Under the a priori assumption (3.3), it holds that 1

N (t) ≤ C[1 + Λ(Q(0))] + C(t 2 + ε)Λ(N (t)), where N (t) is defined in (1.39), the constant C > 0 may depend on a, a ¯, b and ¯b, and Λ(·) is a finite order polynomial. Firstly, we have Lemma 3.2. For s ≥ 4, it holds that ˜ 22 + θ − θ L

s 

∂

α

θ2L2

|α|=1

+

s t 

t ∂

α

θx 2L2 dτ

≤ C + Q (0) + 2

|α|=0 0

Λ(Q(s))ds.

(3.4)

0

Proof. It follows from (1.36)3 that ˜ t + u(θ − θ) ˜ x + ux − κe−εp (eθ (θ − θ) ˜ x )x (θ − θ) =μ ˜ε2 e−εp |ux |2 − uθ˜x + κe−εp (eθ θ˜x )x − θ˜t .

(3.5)

Multiplying (3.5) by θ − θ˜ and integrating over R, it holds that    1 d 2 2 ˜ ˜ ˜ − θ) ˜ x dx |θ − θ| dx + κ a(εp)b(θ)|(θ − θ)x | dx + κεpx eθ−εp (θ − θ)(θ 2 dt ˜ 2 2 + Cux L2 θ − θ ˜ L2 + Cε2 θ − θ ˜ L∞ ux 2 2 ≤ Cux L∞ θ − θ L

L

˜ x L2 )θ − θ ˜ L2 + Cθ − θ ˜ L2 + θ˜x L∞ (uL2 + (θ − θ) ≤ C + Λ(Q(t)).

(3.6)

Integrating (3.6) with respect to time, one obtains that t 2 ˜ (θ − θ)(t) L2 +

t 2 ˜ x (τ )2 2 dτ ≤ C + (θ − θ)(0) ˜ (θ − θ) L L2 +

0

Λ(Q(τ ))dτ. 0

Let θα = ∂ α θ, for 1 ≤ |α| ≤ s. Applying ∂ α to (1.36)3 , one obtains that

(3.7)

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375

∂t θα + u∂x θα + ∂ α ux − κe−εp (eθ ∂x θα )x   ε = −[∂ α , u]θx + μ ˜ε2 ∂ α (e−εp |uεx |2 ) + κ ∂ α (e−εp (eθ θx )x ) − e−εp (eθ ∂x θα )x . (3.8) Multiplying (3.8) by θα and integrating the resulting equation over y, one has that 1 d 2 dt



 |θα |2 dx +

a(εp)b(θ)|∂x θα |2 dx +

1 2



 u∂x (|θα |2 )dx +

θα ∂ α ux dx

  ε   ≤ C  a(εp)b(θ)εpx θα ∂x θα dx + θα L2 [∂ α , u]θx L2 + Cθα L2 ∂ α (e−εp |εuεx |2 )L2       + C  θα ∂ α (e−εp (eθ θx )x ) − e−εp (eθ ∂x θα )x dx. (3.9)

Integrating by parts and using the Cauchy inequality, we obtain that          u∂x (|θα |2 )dx +  θα ∂ α ux dx ≤ ux L∞ θα 2L2 + C∂ α u2L2 + C∂x θα 2L2 ≤ uH 2 θα 2L2 + C∂ α u2L2 + C∂x θα 2L2 ≤ Λ(Q(t)).

(3.10)

We shall estimate the RHS of (3.9). Firstly, we have that      a(εp)b(θ)εpx θα ∂x θα dx ≤ Cεp2H 2 θα 2L2 + C∂x θα 2L2 ≤ Λ(Q(t)).

(3.11)

Notice that [∂ α , u]θx =



Cα,β ∂ β u · ∂ α−β θx ,

(3.12)

1≤|β|≤α

then one gets immediately that [∂ α , u]θx L2 ≤ C

  |β|≤s

    ∂ β uL2 · ∂ γ θx L2 ≤ CΛ(Q(t)).

(3.13)

|γ|≤s−1

A directly calculation shows that ∂ α (e−εp |εuεx |2 )L2 ≤ C∂ α (|εuεx |2 )L2 + Λ(εpHs ) · (εux )2 Hs−1 ε

≤ Cεuεx 2Hs + Λ(εpHs ) · (εux )2 Hs−1 ≤ CΛ(Q(t)).

(3.14)

Noting that ∂ α (e−εp (eθ θx )x ) − e−εp (eθ ∂x θα )x   = Cα,β ∂ β (e−εp ) · ∂ α−β (eθ θx )x + e−εp 1≤|β|≤α

 1≤|β|≤α

Cα,β ∂(eθ ) · ∂ α−β θx

 x

,

376

F. Huang et al. / Advances in Mathematics 319 (2017) 348–395

which yields immediately that ∂ α (e−εp (eθ θx )x ) − e−εp (eθ ∂x θα )x L2   ≤C ∂ β (e−εp ) · ∂ α−β (eθ θxx + eθ |θx |2 )L2 + ∂ β (eθ θx ) · ∂ α−β θx  1≤|β|≤α

 + ∂ β (eθ ) · ∂ α−β θxx    ≤ CΛ(εpHs ) · θxx Hs−1 + Λ(θx Hs ) + CΛ(θx Hs ) · [1 + θxx Hs−1 ] ≤ CΛ(Q(t)).

(3.15)

Substituting (3.10), (3.11), (3.13), (3.14) and (3.15) into (3.9), one obtains that   1 d |θα |2 dx + a(εp)b(θ)|∂x θα |2 dx ≤ CΛ(Q(t)). (3.16) 2 dt Integrating (3.16) with respect to time, one gets that t θα 2L2

t ∂x θα (τ ) dτ ≤ 2

+

Cθα (0)2L2

0

+C

Λ(Q(τ ))dτ.

(3.17)

0

Then, from (3.17) and (3.7), we obtain (3.4). Thus the proof of Lemma 3.2 is completed. 2 Lemma 3.3. For s ≥ 4, it holds that t (εp, εu)(t)2Hs

t εux (τ )2Hs dτ

+

≤ CQ (0) + C

Λ(Q(τ )) · [1 + S(τ )]dτ.

2

0

(3.18)

0

Proof. For simplicity, we denote p˘ = εp, u ˘ = εu, and (˘ pα , u ˘α ) = ∂ α (˘ p, u ˘), 0 ≤ |α| ≤ s.

(3.19)

Then the functions (˘ p, u ˘) satisfy ⎧ ⎨∂t p˘ + u˘ px + (2u − a(˘ p)b(θ)θx )x = a(˘ p)|˘ ux |2 + a(˘ p)b(θ)θx · p˘x , ⎩b(−θ)[∂ u ux ] + p x = μ ˜a(˘ p)˘ uxx . t ˘ + u˘ Applying ∂ α to (3.20), one gets that ⎧ ⎨∂t p˘α + u∂x p˘α = h1 + h2 + h3 + h4 , ⎩b(−θ)[∂ u ˘ α ] + ∂ α px − μ ˜a(˘ p)∂xx u ˘α = h5 + h6 + h7 , t ˘α + u∂x u

(3.20)

(3.21)

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377

where   ⎧ α α ⎪ = −[∂ , u]∂ p ˘ , h = −∂ ∂ 2u − a(˘ p )b(θ)θ , h 1 x 2 x x ⎪ ⎪ ⎨     p)|˘ ux |2 , h4 = ∂ α a(˘ p)b(θ)θx · p˘x , h3 = ∂ α a(˘ ⎪ ⎪ ⎪ ⎩ ˘, h6 = −[∂ α , b(−θ)u]∂x u ˘ , h7 = μ ˜[∂ α , a(˘ p)]˘ uxx . h5 = −[∂ α , b(−θ)]∂t u Multiplying (3.21)1 by p˘α , one has that    1 d 1 |˘ pα |2 dx = ux |˘ pα |2 dx + (h1 + h2 + h3 + h4 )˘ pα dx. 2 dt 2

(3.22)

(3.23)

It is straightforward to imply that     pα |2 dx ≤ ux L∞ ˘ pα 2L2 ≤ CΛ(Q(t)).  ux |˘

(3.24)

A directly calculation shows that (h1 , h3 , h4 )L2 ≤ CΛ(Q(t)), which yields immediately that     pα dx ≤ CΛ(Q(t)).  (h1 + h3 + h4 )˘

(3.25)

Noting that h2 L2 ≤ C[∂ α ux L2 + ∂ α ∂xx θL2 + Λ(Q(t))] ≤ C[S(t) + Λ(Q(t))],

(3.26)

we obtain that     pα L2 · [S(t) + Λ(Q(t))].  h2 p˘α dx ≤ C˘

(3.27)

Substituting (3.24), (3.25) and (3.27) into (3.23), then integrating the resulting inequality with respect to time, then we get that t ˘ pα  ≤ ˘ pα (0) + C 2

Λ(Q(τ )) · [1 + S(τ )]dτ.

2

(3.28)

0

On the other hand, multiplying (3.21)2 by u ˘α and integrating the resulting equation, one obtains that    1 d 2 b(−θ)|˘ uα | dx − μ ˜ a(˘ p)˘ uα ∂xx u ˘α dx ≤ (h5 + h6 + h7 )˘ uα dx, (3.29) 2 dt

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378

where we have used the facts         2 uα | dx +  b(−θ)u(|˘ uα |2 )x dx ≤ CΛ(Q(t)),  ∂t b(−θ)|˘

(3.30)

    ˘α dx ≤ ∂ α pL2 · ∂x u ˘α L2 ≤ CΛ(Q(t)).  ∂ α px · u

(3.31)

and

For the second term on the left hand side of (3.29), it follows from integrating by parts that  −˜ μ

 a(˘ p)˘ uα ∂xx u ˘α dx = μ ˜ ≥

3 μ ˜ 4

 a(˘ p)|∂x u ˘α | dx + μ ˜ 2

∂x a(˘ p)˘ uα ∂x u ˘α dx

 a(˘ p)|∂x u ˘α |2 dx − CΛ(Q(t)).

(3.32)

Now we estimate the RHS of (3.29). Noting h5 L2 = [∂ α , b(−θ)]∂t u ˘L2 = [∂ α , b(−θ)](ε∂t )uL2 ≤ CΛ(Q(t))uHs ≤ CΛ(Q(t)), and h6 L2 + h7 L2 ≤ CΛ(Q(t)) + CΛ(Q(t))εux Hs ≤ CΛ(Q(t)), which, together with Holder inequality, yield that     uα dx ≤ Λ(Q(t)).  (h5 + h6 + h7 )˘

(3.33)

Substituting (3.33)–(3.30) into (3.29) and integrating the resulting inequality with respect to time, one obtains that t εu(t)2Hs

t εu(τ )2Hs dτ

+ 0

≤

Cεu(0)2Hs

+

Λ(Q(τ ))dτ.

(3.34)

0

Combining (3.34) and (3.28), one proves (3.18). Thus the proof of Lemma 3.3 is completed. 2 The following Lemma is devoted to the highest order derivative estimates ∂ α (εp, εu, θ)(t)2L2 , |α| = s + 1.

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Lemma 3.4. For s ≥ 4, it holds that 

∂

α

(εp, εu, θ)(t)2L2

+

|α|=s+1

t  

∂ α (εux , θx )(τ )2L2 dτ

|α|=s+1 0

t

(3.35)

[1 + S(τ )]Λ(Q(τ ))dτ.

≤ CQ2 (0) + C 0

Proof. Similar to [2,25], we set (ˇ p, u ˇ) = (εp − θ, εu). Then, from (1.36), it is easy to check that (ˇ p, u ˇ, θ) satisfy ⎧ ⎪ px + 1ε u ˇx = 0, ∂t pˇ + uˇ ⎪ ⎪ ⎨ b(−θ)[∂t u ˇ + uˇ ux ] + 1ε (ˇ px + θx ) = μ ˜a(εp)ˇ uxx , ⎪ ⎪ ⎪ ⎩∂ θ + uθ + 1 u 2 ˜ε a(εp)|ux |2 . t x ε ˇx = a(εp)(b(θ)θx )x + μ

(3.36)

(3.37)

Let α be a multi-index with |α| = s + 1 and denote (ˇ pα , u ˇα , θα ) = ∂ α (ˇ p, u ˇ, θ).

(3.38)

Applying ∂ α to (3.37), one gets that ⎧ ⎪ ˇα = h8 , ∂t pˇα + u∂x pˇα + 1ε ∂x u ⎪ ⎪ ⎨

ˇα + u∂x u ˇα ] + 1ε (∂x pˇα + ∂x θα ) = μ ˜a(εp)∂xx u ˇα + h9 , b(−θ)[∂t u ⎪ ⎪ ⎪ ⎩∂ θ + u∂ θ + 1 ∂ u t α x α ε x ˇα = a(εp)(b(θ)∂x θα )x + h10 ,

(3.39)

where h8 = −[∂ α , u]ˇ px , ˜a(εp)]∂xx u ˇ, h9 = −[∂ α , b(−θ)]∂t (εu) − [∂ α , b(−θ)u]∂x (εu) + [∂ α , μ     h10 = −[∂ α , u]θx + μ ˜∂ α ε2 a(εp)|ux |2 + ∂ α (a(εp)(b(θ)θx )x ) − a(εp)(b(θ)∂x θα )x . !  Considering ˇα + (3.39)1 × θα dx and integrating by parts (3.39)1 × pˇα + (3.39)2 × u the resulting equation, one can obtain   3 1 d 2 2 2 |ˇ pα | + b(−θ)|ˇ μ ˜a(εp)|∂x u uα | + |θα | dx + ˇα |2 + a(εp)b(θ)|∂x θα |2 dx 2 dt 4  ˇα + h10 θα )dx, (3.40) ≤ CΛ(Q(t)) + (h8 pˇα + h9 u

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where we have used the following estimates             pα ∂x pˇα dx +  ∂t b(−θ) · |ˇ uα |2 dx +  b(εp)uˇ u α ∂x u ˇα dx  uˇ     +  uθα ∂x θα dx ≤ Λ(Q(t)), 





ˇα u ˇα dx = − a(εp)∂xx u

a(εp)|∂x u ˇα | dx − ∂x a(εp)∂x u ˇα u ˇα dx  3 a(εp)|∂x u ˇα |2 dx + CΛ(Q(t)), ≤− 4 2



 a(εp)(b(θ)∂x θα )x θα dx = − ≤−

(3.41)

 a(εp)b(θ)|∂x θα | dx − 2

3 4

(3.42)

∂x a(εp) · b(θ)θα ∂x θα dx

 a(εp)b(θ)|∂x u ˇα |2 dx + CΛ(Q(t)),

(3.43)

and the fact that  ˇα pˇα + ∂x pˇα u ˇα + u ˇα ∂x θα + ∂x u ˇα θα ) dx = 0. (∂x u

(3.44)

It remains to estimate the RHS of (3.40). Firstly, it follows from (1.36)2 that ε∂t u = −u∂x (εu) − b(θ)∂x p + μ ˜a(εp)b(θ)∂xx (εu), which yields immediately that   ε∂t uHs ≤ CΛ(Q(t)) 1 + ∂x (εu)Hs + ∂x pHs + ∂xx (εu)Hs   ≤ CΛ(Q(t)) 1 + S(t) .

(3.45)

Using (3.45), one gets that h8 L2 = [∂ α , u](εpx − θx )L2 ≤ Cεpx − θx Hs uHs+1   ≤ Λ(Q(t)) ε∂t uHs + ux Hs + uHs   ≤ CΛ(Q(t)) 1 + S(t) , which, together with Holder inequality, implies that        h8 pˇα dx ≤ CΛ(Q(t)) 1 + S(t) .

(3.46)

Next, we shall estimate the terms involving h9 . It is straightforward to imply that

F. Huang et al. / Advances in Mathematics 319 (2017) 348–395

[∂ α , b(−θ)u]∂x (εu)L2 + [∂ α , μ ˜a(εp)]∂xx u ˇ  L2     ≤ CΛ(Q(t)) 1 + εuHs+1 + εuxx Hs ≤ CΛ(Q(t)) 1 + S(t) ,

381

(3.47)

and   [∂ α , b(−θ)]∂t (εu)L2 ≤ CΛ(Q(t))[1 + ε∂t uHs ] ≤ CΛ(Q(t)) 1 + S(t) , (3.48) where we have used (3.45) in the last inequality of (3.48). Then, it follows from (3.47) and (3.48) that       ˇα dx ≤ CΛ(Q(t)) 1 + S(t) . (3.49)  h9 u Finally, we estimate the terms involving h10 . Similarly, we have that   h10 L2 ≤ CΛ(Q(t)) 1 + S(t) , which yields that        h10 θα dx ≤ CΛ(Q(t)) 1 + S(t) .

(3.50)

Substituting (3.50), (3.49) and (3.46) into (3.40) and integrating the resulting inequality with respect to time, one gets (3.35). Thus the proof of Lemma 3.4 is completed. 2 To establish the estimates for (p, u)Hs , we first control the term (ε∂t )s (p, u) which plays key role in the estimates for (p, u)Hs . We start with a L2 -estimate for the following linearized equations around a given state (p, u, θ) ⎧ ⎪ ˜εa(εp)ux ulx + κa(εp)b(θ)px θxl + f1 , pl + u · plx + 1ε (2ul − κa(εp)b(θ)θxl )x = μ ⎪ ⎪ ⎨ t ˜a(εp)ulxx + f2 , b(−θ)[ult + u · ulx ] + 1ε plx = μ ⎪ ⎪ ⎪ ⎩θl + uθl + ul = κa(εp)(b(θ)θl ) + μ ˜ε2 a(εp)ux ulx + f3 , t x x x x where we fi , i = 1, 2, 3 are source terms. Lemma 3.5. Let (pl , ul , θl ) be the solution of (3.51) and assume that 0<

1 1 a ≤ a(εp) ≤ 2¯ a, and 0 < b ≤ b(θ) ≤ 2¯b, on t ∈ [0, T ], 2 2

then it holds that, for 0 < t ≤ T , t (p , u l

l

)(t)2L2

ulx (τ )2L2 dτ

+ 0

(3.51)

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≤ C(pl , ul )(0)2L2 + C sup θxl (τ )2L2 0≤τ ≤t

" t

t l (εulx , θxx )2L2 dτ

+ CΛ(R0 )

+



(p , u

l

l

, θxl )2L2 dτ

0

t l

(p , u

+

0

t l

, θxl )2L2 dτ

0

12 ·

t f1 , f2 2L2 dτ

f3 2L2 dτ

+ 0

12

# ,

(3.52)

0

. where R0 = supτ ∈[0,T ] {∂t θ(τ )L∞ + (p, u, θ)(τ )W 1,∞ }. Proof. Following [25], we define u := 2ul − κa(εp)b(θ)θxl .

(3.53)

Then, it is straightforward to check that pl solves 1 μ ˜ plt + u · plx + ux = μ ˜εa(εp)ux ux + εa(εp)ux (κa(εp)b(θ)θxl )x + κa(εp)b(θ)px · θxl + f1 . ε 2 (3.54) Consider 12 a(εp) · ∂x (3.51)3 , one can get that   1 b(−θ) ∂t (a(εp)b(θ)θxl ) + u · (a(εp)b(θ)θxl )x 2   κ 1 = b(−θ) ∂t (a(εp)b(θ))θxl + u · (a(εp)b(θ))x θxl − a(εp)[a(εp)b(θ)θxl ]xx 2 4 μ ˜ 2 μ ˜ 2 1 + ε a(εp)[a(εp)ux ux ]x + ε a(εp)[a(εp)ux · (a(εp)b(θ)θxl )x ]x − a(εp)ux θxl 4 4 2 1 1 1 + a(εp)[κa(εp)(b(θ)θxl )x ]x − a(εp)uxx + a(εp) · ∂x f3 , 2 4 2 which, together with (3.51)2 , yields that   1 1 1 1 b(−θ) ∂t u + u · ux + plx − a(εp)uxx − μ ˜a(εp)uxx 2 ε 4 2   κ 1 = − b(−θ) ∂t (a(εp)b(θ))θxl + u · (a(εp)b(θ))x θxl + a(εp)[a(εp)b(θ)θxl ]xx 2 4 μ ˜ 2 μ ˜ 2 1 − ε a(εp)[a(εp)ux ux ]x − ε a(εp)[a(εp)ux · (a(εp)b(θ)θxl )x ]x + a(εp)ux θxl 4 4 2 1 μ ˜ 1 − a(εp)[κa(εp)(a(θ)θxl )x ]x + a(εp) · [κa(εp)b(θ)θxl ]xx + f2 − a(εp) · ∂x f3 2 2 2 1 =: g + f2 − a(εp) · ∂x f3 . (3.55) 2

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Multiplying (3.54) by pl , (3.55) by u, adding them together and integrating the resulting equation, we get 

 1 1 d 1 l2 1 2 |p | + b(−θ)|u| dx + ( + μ ˜)a(εp)|ux |2 dx dt 2 4 4 2    1 1 1 1 l 2 2 ux |p | dx + [∂t b(−θ) + ∂x (b(−θ)u)]|u| dx + ( + μ ˜)εa(εp)px ux udx = 2 4 4 2    μ ˜ εa(εp)ux (κa(εp)b(θ)θxl )x pl dx + κ a(εp)b(θ)px · θxl pl dx + μ ˜εa(εp)ux ux pl dx + 2    1 a(εp)∂x f3 · udx + gudx + (f1 pl + f2 u)dx − 2  1 1 1 l ( + μ ˜)a(εp)|ux |2 dx + CΛ(R0 )(pl , ul , θxl , θxx )2 + f3 2L2 ≤ 8 4 2   + (f1 pl + f2 u)dx + gudx. (3.56) It is easy to note that   1 1 1   l ˜)a(εp)|ux |2 dx + CΛ(R0 )(pl , ul , θxl , θxx )2L2 . ( + μ  gudx ≤ 8 4 2

(3.57)

Substituting (3.57) into (3.56) and integrating the resulting inequality with respect to time, one obtains (3.52). Therefore we complete the proof of Lemma 3.5. 2 Next we shall use Lemma 3.5 to estimate (ε∂t )k (p, u)L2 for 1 ≤ k ≤ s. Lemma 3.6. For s ≥ 4 and 0 ≤ k ≤ s, it holds that t (ε∂t )

k

(p, u)(t)2L2

(ε∂t )k ux (τ )2L2 dτ

+ 0

≤ C(ε∂t )

k

(p, u)(0)2L2

1

+ C sup θx (τ )2Hk + Ct 2 Λ(N (t)) 0≤τ ≤t

t θxx (τ )2Hk dτ,

+ CΛ(R)

(3.58)

0

  where R(t) = sup0≤τ ≤t ∂t θ(τ )L∞ + (p, u, θ)(τ )W 1,∞ . Proof. For simplicity, we set (pk , uk , θk ) = (ε∂t )k (p, u, θ), 0 ≤ k ≤ s. Applying (ε∂t )k to (1.36), one has that

(3.59)

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⎧ 1 ⎪ ˜εa(εp)ux ukx + κa(εp)b(θ)px · θkx + fk1 , ⎪ ⎪pkt + u · pkx + ε (2uk − κa(εp)b(θ)θkx )x = μ ⎨ 1 (3.60) ˜a(εp)ukxx + fk2 , b(−θ)[ukt + u · ukx ] + ε pkx = μ ⎪ ⎪ ⎪ ⎩θ + uθ + u = κa(εp)(b(θ)θ ) + μ ˜ε2 a(εp)ux ukx + fk3 , kt kx kx kx x where ⎧   ⎪ ˜εa(εp)ux ]ux ⎪ fk1 = −[(ε∂t )k , u]px + κ 1ε [(ε∂t )k , a(εp)b(θ)]θx + ε[(ε∂t )k , μ ⎪ ⎪ x ⎪ ⎪ ⎪ ⎪ ⎪ + [(ε∂t )k , κa(εp)b(θ)px ]θx , ⎪ ⎨ fk2 = −[(ε∂t )k , b(−θ)]ut − [(ε∂t )k , b(−θ)u]ux + μ ˜[(ε∂t )k , a(εp)]uxx , ⎪ ⎪ ⎪ ⎪ ⎪ fk3 = −[(ε∂t )k , u]θx + μ ˜ε[(ε∂t )k , a(εp)ux ]ux + κ[(ε∂t )k , a(εp)](b(θ)θx )x ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ + κa(εp)([(ε∂t )k , b(θ)]θx )x .

(3.61)

Applying Lemma 3.5 to equations (3.60), we obtain t (ε∂t )

k

(p, u)(t)2L2

(ε∂t )k ux 2L2 dτ

+ 0

" t ≤ C(ε∂t )k (p, u)(0)2L2 + C sup θx (τ )2Hk + CΛ(R) 0≤τ ≤t

t

t (pk , uk , (θk )x )2L2 dτ

+ 0

fk3 2L2 dτ

+

(ε∂t )k θx x(τ )2L2 dτ 0

+ t Λ(Q(t)) 1 2

0

t (fk1 , fk2 )2L2 dτ

12

# .

0

(3.62) It remains to estimate the terms involving (fk1 , fk2 , fk3 ). We need only to estimate the case for k ≥ 1 since (f01 , f02 , f03 ) = (0, 0, 0). For the singular term of fk1 , we first notice that 

[(ε∂t )k , a(εp)b(θ)]θx

 x

= [(ε∂t )k , a(εp)b(θ)]θxx + [(ε∂t )k , ∂x (a(εp)b(θ))]θx ,

(3.63)

then a careful calculation gives that 1 [(ε∂t )k , a(εp)b(θ)]θxx L2 = [(ε∂t )k−1 ∂t , a(εp)b(θ)]θxx L2 ε ≤ [(ε∂t )k−1 , a(εp)b(θ)]∂t θxx L2 + [(ε∂t )k−1 , ∂t (a(εp)b(θ))]θxx L2 ≤ CΛ(Q(t))[1 + ∂t θx Hs−1 + ∂t θHs−1 ], and

(3.64)

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1 [(ε∂t )k , ∂x (a(εp)b(θ))]θx L2 = [(ε∂t )k−1 ∂t , ∂x (a(εp)b(θ))]θx L2 ε ≤ [(ε∂t )k−1 ∂t , ∂xt (a(εp)b(θ))]θx L2 + [(ε∂t )k−1 , ∂x (a(εp)b(θ))]θxt L2 ≤ CΛ(Q(t))[1 + ∂t θx Hs−1 + px Hs ].

(3.65)

For the remain terms of fk1 , it is straightforward to obtain [(ε∂t )k , u]px L2 + ε[(ε∂t )k , μ ˜εa(εp)ux ]ux L2 + [(ε∂t )k , κa(εp)b(θ)px ]θx L2 ≤ CΛ(Q(t))[1 + px Hs ].

(3.66)

Combining (3.63)–(3.66), one gets that fk1 L2 ≤ CΛ(Q(t))[1 + ∂t θx Hs−1 + px Hs ].

(3.67)

For the estimate of fk2 L2 , it is noted that [(ε∂t )k , b(−θ)]ut =

k 

Ck,i (ε∂t )i b(−θ) · (ε∂t )k−i ut

i=1

=

k 

Ck,i (ε∂t )i−1 ∂t b(−θ) · (ε∂t )k−i+1 u,

i=1

which yields immediately that [(ε∂t )k , b(−θ)]ut L2 ≤ CΛ(Q(t))[1 + ∂t θHk−1 + Λ(θt Hk−2 )],

(3.68)

in which θt Hk−2 does not appear when k − 2 < 0. For the second and third terms of fk2 , it is straightforward to obtain that [(ε∂t )k , b(−θ)u]ux L2 + ˜ μ[(ε∂t )k , a(εp)]uxx L2 ≤ CΛ(Q(t)).

(3.69)

Then it follows from (3.68) and (3.69) that fk2 L2 ≤ CΛ(Q(t))[1 + ∂t θHk−1 + Λ(θt Hk−2 )].

(3.70)

Finally, a direct calculation yields that fk3 L2 ≤ CΛ(Q(t)).

(3.71)

Note that both (3.67) and (3.70) contain some norms of ∂t θ, which are not included in Q(t). To bound ∂t θ, we use (1.36)3 to obtain   θt Hs−1 ≤ CΛ(Q(t)), and θtx Hs−1 ≤ CΛ(Q(t)) 1 + ux Hs + θxx Hs . (3.72)

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Then, combining (3.72), (3.70) and (3.67), one gets that (fk1 , fk2 )L2 ≤ CΛ(Q(t))[1 + (px , ux , θxx )Hs ].

(3.73)

Substituting (3.71) and (3.73) into (3.62), one proves (3.58). Therefore, the proof of Lemma 3.6 is completed. 2 From Lemma 3.2, Lemma 3.4 and Lemma 3.6, we have the following corollary: Corollary 3.7. For s ≥ 4 and 0 ≤ k ≤ s − 1, it holds that t (ε∂t )

k

(p, u)(t)2L2

+

  1 (ε∂t )k ux (τ )2L2 dτ ≤ C 1 + Q2 (0) + Ct 2 Λ(N (t)),

(3.74)

0

and t (ε∂t )

s

(p, u)(t)2L2

(ε∂t )s ux (τ )2L2 dτ

+ 0

  1 ≤ C 1 + Λ(Q(0)) + Ct 2 Λ(N (t)) + CΛ(R(t)).

(3.75)

where R(t) is defined in Lemma 3.6. We now use Corollary 3.7 to estimate (p, u)Hs . It follows from (1.36) that ⎧ ⎨2ux = −ε∂t p − εupx + (κa(εp)b(θ)θx )x + μ ˜a(εp)|εux |2 + κa(εp)b(θ)εpx · θx , ⎩p = −b(−θ)ε∂ u − εb(−θ)uu + μ ˜a(εp)εuxx . x t x Lemma 3.8. It holds, for s ≥ 4, that   1 (p, u)Hs ≤ C 1 + Λ(Q(0)) + C(t 2 + ε)Λ(N (t)).

(3.76)

(3.77)

Proof. It follows from (3.76)1 , Lemma 3.2–Lemma 3.4 and Corollary 3.7 that   ux L2 ≤ C (ε∂t )pL2 + εuH 1 px L2 + θxx L2 + Λ((εp, εux )Hs + θx Hs−1 )   1 ≤ C 1 + Λ(Q(0)) + C(t 2 + ε)Λ(N (t)). (3.78) Similarly, one has, for 0 ≤ k ≤ s − 2, that   (ε∂t )k ux L2 ≤ C (ε∂t )k+1 pL2 + εΛ(Q) + Λ((εp, εu)Hs + (ε∂t θ, θx , θxx )Hs−1 )   1 ≤ C 1 + Λ(Q(0)) + C(t 2 + ε)Λ(N (t)), (3.79)

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and   (ε∂t )s−1 ux L2 ≤ C (ε∂t )s pL2 + εΛ(Q) + Λ((εp, εu)Hs + (ε∂t θ, θx , θxx )Hs−1 )   1 ≤ C 1 + Λ(Q(0)) + C(t 2 + ε)Λ(N (t)) + CΛ(R(t)). (3.80) For px , it follows from (3.76)2 , Lemma 3.2–Lemma 3.4 and Corollary 3.7, for 0 ≤ k ≤ s − 2, that   (ε∂t )k px L2 ≤ C (ε∂t )k+1 uL2 + εΛ(Q) + Λ((εp, εu)Hs + (ε∂t θ, εuxx )Hs−1 )   1 ≤ C 1 + Λ(Q(0)) + C(t 2 + ε)Λ(N (t)), (3.81) and   (ε∂t )s−1 px L2 ≤ C (ε∂t )s uL2 + εΛ(Q) + Λ((εp, εu)Hs + (ε∂t θ, εuxx )Hs−1 )   1 ≤ C 1 + Λ(Q(0)) + C(t 2 + ε)Λ(N (t)) + CΛ(R(t)). (3.82) Using the system (3.76) and (3.79), (3.81), one obtains, for 0 ≤ k ≤ s − 3 that   1 (ε∂t )k (pxx , uxx )L2 ≤ C 1 + Λ(Q(0)) + C(t 2 + ε)Λ(N (t)).

(3.83)

Similarly, one can get that   1 (p, u)Hs−1 ≤ C 1 + Λ(Q(0)) + C(t 2 + ε)Λ(N (t)).

(3.84)

Then it follows from (1.36)3 , (3.4) and (3.84) that   1 R(t) ≤ C 1 + Λ(Q(0)) + C(t 2 + ε)Λ(N (t)).

(3.85)

Combining (3.75) (3.76), (3.80), (3.82), (3.84), (3.85), and using the same argument as in (3.84), one can get that   1 (p, u)Hs ≤ C 1 + Λ(Q(0)) + C(t 2 + ε)Λ(N (t)).

(3.86)

Therefore the proof of Lemma 3.8 is completed. 2 Lemma 3.9. It holds, for s ≥ 4, that t 0

  1 (px , ux )(τ )2Hs dτ ≤ C 1 + Λ(Q(0)) + C(t 2 + ε)Λ(N (t)).

(3.87)

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Proof. Firstly, it follows from (3.75) and (3.85) that t

  1 (ε∂t )s ux (τ )2L2 dτ ≤ C 1 + Λ(Q(0)) + C(t 2 + ε)Λ(N (t)).

(3.88)

0

Using (3.88) and (3.76)1 , one can obtain that t (ε∂t )s+1 p(τ )2L2 dτ 0

t ≤C

  1 (ε∂t )s ux (τ )2L2 dτ + C 1 + Λ(Q(0)) + C(t 2 + ε)Λ(N (t))

0

  1 ≤ C 1 + Λ(Q(0)) + C(t 2 + ε)Λ(N (t)).

(3.89)

On the other hand, it follows from (3.76)2 that   (ε∂t )s px = −εb(−θ)(ε∂t )s ut + (ε∂t )s − εb(−θ)uux + μ ˜a(εp)εuxx .

(3.90)

Multiplying (3.90) by (ε∂t )s px and integrating the resulting equation yield that  (ε∂t )s px 2L2 = −ε  + ≤−

d dt 

b(−θ)(ε∂t )s ut · (ε∂t )s px dx   (ε∂t )s px · (ε∂t )s − εb(−θ)uux + μ ˜a(εp)εuxx dx 

 b(−θ)(ε∂t )s u · (ε∂t )s (εpx )dx +

b(−θ)(ε∂t )s u · (ε∂t )s+1 px dx

1 ∂t b(−θ)(ε∂t )s u · (ε∂t )s (εp)x dx + (ε∂t )s px 2L2 8    2   + (ε∂t )s − εb(−θ)uux + μ ˜a(εp)εuxx  dx. +

(3.91)

Integrating (3.91) with respect to time, using (3.88), (3.89), Lemma 3.2–Lemma 3.4 and Lemma 3.8, one immediately gets that t

  1 (ε∂t )s px (τ )2L2 dτ ≤ C 1 + Λ(Q(0)) + C(t 2 + ε)Λ(N (t)).

0

Then, from (3.76), (3.89) and (3.92), one obtains that

(3.92)

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t (ε∂t )s−1 (pxx , uxx )(τ )2L2 dτ 0

t ≤C

  1 (ε∂t )s (px , ux )(τ )2L2 dτ + C 1 + Λ(Q(0)) + C(t 2 + ε)Λ(N (t))

0



 1 ≤ C 1 + Λ(Q(0)) + C(t 2 + ε)Λ(N (t)).

(3.93)

In the same way, one can prove that t

  1 (px , ux )(τ )2Hs dτ ≤ C 1 + Λ(Q(0)) + C(t 2 + ε)Λ(N (t)).

0

Therefore the proof of Lemma 3.9 is completed. 2 Proof of Proposition 3.1. Proposition 3.1 follows immediately from Lemma 3.2–Lemma 3.4, Lemma 3.8 and Lemma 3.9. 2 Proof of Theorem 1.6. Using Proposition 3.1, one can prove Theorem 1.6 by combining the local existence theorem and the bootstrap arguments, and thus close the a priori assumption (3.3). The details are omitted here for simplicity of presentation. 2 3.2. Low Mach limit In this subsection, we will prove Theorem 1.9 with a modified compactness argument, which was introduced by Métivier and Schochet in [35], see also the extensions in [1,2,31]. Proof of Theorem 1.9. Firstly, it follows from Theorem 1.6 that ˜ )H s+1 < ∞. sup (pε , uε )(τ )H s + sup (θε − θ)(τ

τ ∈[0,T0 ]

(3.94)

τ ∈[0,T0 ]

Then extracting a subsequence, it holds that (pε , uε )  (p, u) as ε → 0 weak-∗ in L∞ (0, T0 ; H s (R)), θε − θ˜  θ¯ − θ˜ as ε → 0 weak-∗ in L∞ (0, T0 ; H s+1 (R)). It follows from the equation of θε and (3.94) that θtε ∈ L∞ (0, T0 ; H s−2 (R)), which, together with Aubin–Lions lemma, yields that the functions θε converge (possibly s +1 after extracting a subsequence) to θ¯ strongly in C([0, T0 ]; Hloc (R)) for all s < s.

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To obtain the limiting system (1.37), we need to show the strong convergence of s (p , uε ) in L2 (0, T0 ; Hloc (R)) for s < s. To this end, we will show pε and (2uε − ε ε θ −εp ε κe θx )x converge strongly to 0 as ε → 0. In fact we rewrite (1.36)1 and (1.36)2 as, ε

εpεt + (2uε − κeθ

ε

−εpε ε θx )x

= εf ε ,

(3.95)

and εe−θ uεt + pεx = εg ε . ε

(3.96)

It follows from (1.42) that f ε and g ε are uniformly bounded in C([0, T0 ]; H s−1 (R)). Noticing pε are uniformly bounded in L∞ (0, T0 , L∞ (R)), passing the weak limit in (3.95) ¯ and (3.96) leads to (p)x = 0 and (2u − κeθ θ¯x )x = 0 in the distribution sense. On the other hand, by taking the limit of (1.36)3 , we can get that θ¯ satisfies ¯

κeθ ¯ θ¯t = θxx , 2

(3.97)

¯ 0) = θin (x). θ(x,

(3.98)

with the initial data,

From the maximum principle and energy method, one can show the existence and unique¯ as in [2], we define: ness of smooth solution of (3.97), (3.98). To get the spatial decay of θ, H = x1+σ (θ¯ − θ+ ), which satisfies ¯

Ht =

¯

¯

κeθ κ(1 + σ)eθ κ(1 + σ)(2 + δ)eθ Hxx − Hx + H. 2 x x2

By the energy estimates and (1.42), we can obtain ˜ H 2 + 1] ≤ C[1 + Λ(C0 )] HL∞ (0,T0 ;H 1 [1,∞)) ≤ C[H(0)H 1 [1,∞) + θ¯ − θ which, together with Sobolev embedding, yields that ¯ t) − θ+ | ≤ Cx−1−σ , as x ∈ [1, +∞). |θ(x,

(3.99)

To obtain the strong convergence of uε and pε , we need the following Proposition 3.10 which will be proved in the end of this section. Proposition 3.10. Let (1.42) and (3.99) hold, then pε and (2uε − κeθ −εp θxε )x converge s −1 s to 0 strongly in L2 (0, T0 ; Hloc (R)) and L2 (0, T0 ; Hloc (R)) for s < s, respectively. ε

ε

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If Proposition 3.10 holds, passing the limit in the equations (1.36) for (pε , uε , θε ), one proves that the limit (0, u, θ) solves (1.37) in the sense of distribution. ¯ satisfies On the other hand, following the arguments in [35], one can obtain that (¯ u, θ) the initial condition ¯ t=0 = (win , θin ) (¯ u, θ)|

(3.100)

where win is determined by win = 12 κeθin (θin )x . Moreover one can get the uniqueness of solutions to the limit system (1.37) with initial data (3.100) by the energy method and then the above conclusions hold for the whole sequence (pε , uε , θε ). The proof of Theorem 1.9 is completed. 2 ε

Proof of Proposition 3.10. Applying ε∂t to (3.95) and ∂x to eθ × (3.96), we obtain that ε 1 ε2 pεtt − (eθ pεx )x = εF ε (pε , uε , θε ), 2

(3.101)

where F ε (pε , uε , θε ) is a smooth function. From (1.42), εF ε (pε , uε , θε ) converges to 0 strongly in L2 (0, T0 ; L2 (R)) as ε → 0. We now recall the convergence lemma due to Métivier and Schochet [35] in Rd with d = 1, 2, 3. Lemma 3.11. Let T > 0, v ε be a bounded sequence in C([0, T ], H 2 (Rd )) and ε∂t v ε are bounded in L2 (0, T ; L2 (Rd )) satisfying ε2 ∂t (aε ∂t v ε ) − ∇ · (bε ∇v ε ) = cε ,

(3.102)

where cε converges to 0 strongly in L2 (0, T ; L2 (Rd )). Assume further that, for some k > k (Rd )) 1 + d/2, the coefficients (aε , bε ) are positive and uniformly bounded in C([0, T ]; Hloc k (Rd )) to (a, b) satisfying and converge in C([0, T ]; Hloc for all τ ∈ R, the kernel of aτ 2 + ∇ · (b∇) in L2 (Rd ) is reduced to {0}.

(3.103)

Then the sequence v ε converges to 0 strongly in L2 (0, T ; L2loc (Rd )). We introduce the following condition to verify (3.103). Condition A. The functions (a, b) are positive bounded and satisfy |a(x, t) − a+ | ≤ Ca r(x),

|b(x, t) − b+ | ≤ Cb r(x), as x ∈ [1, +∞),

where r(x) ∈ L1 ([1, +∞)) is a non-negative function, and a+ , b+ , Ca , Cb are some positive constants.

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It is obvious that if v ∈ L2 (R) satisfies aτ 2 v + ∂x (b∂x v) = 0,

(3.104)

then v ∈ H 1 (R) and 

 2

b(∂x v) dx = τ R

2

av 2 dx, R

which implies v ≡ 0 when τ = 0. When τ = 0, we assume τ = 1 without loss of generality. Let w := b∂x v, then (3.104) becomes d dx



 v w

 =

 0 h −a 0

 v w

,

(3.105)

where h := 1/b has similar properties of b. From v ∈ H 1 (R), v, w → 0 as |x| → ∞. For further analysis, we need the following Lemma 3.12 which can be verified directly by the energy method. The details are omitted here. Lemma 3.12. Let A(x) be a smooth bounded function on x ∈ R+ . Assume U (0) = 0 with U ∈ Cd . Then U ≡ 0.

d dx U

= A(x)U ,

We shall first prove (v, w)T (x) = 0 for x ∈ [1, +∞). It is noted that (a, h) → (a+ , h+ ) as x → +∞. Then we rewrite (3.105) as d dx



 v w

 +

0 a+

−h+ 0



 v w



0 h − h+ = −(a − a+ ) 0    ˜ 0 h v =: . −˜ a 0 w



A direct calculation shows that B = Q−1 ΛQ with     √ a+ h+ i 0 a+ 0 −h+ , Λ= B= and Q = √  a+ 0 0 − a+ h+ i a+

 v w (3.106)

 h+ i .  − h+ i (3.107) 

It follows from (3.106) and (3.107) that   √  √   a+ v + h+ iw a + h+ i 0 a+ v + h+ wi d +    √ dx √a+ v − h+ iw 0 − a+ h+ i a+ v − h+ wi   √   ˜ 0 h a+ h+ i v = √ , (3.108)  w −˜ a 0 a+ − h+ i

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which yields immediately that 

 a+ v + h+ wi  

√ e− a+ h+ ix a+ v − h+ wi    √   0 a+ e a+ h+ ix h+ ie a+ h+ ix   = √  − a+ h+ ix − a+ h+ ix −˜ a a+ e − h+ ie

d dx



e

 a+ h+ ix

√

˜ h



 v w

0

.

Then, one has that d dx



    √

 √

 e a+ h+ ix a+ v + h+ wi a+ v + h+ wi ˜     √

=B √

, a+ v − h+ wi a+ v − h+ wi e− a+ h+ ix e− a+ h+ ix e



a+ h+ ix

where  √ ˜ := B

√

a+ e



a+ h+ ix



   ˜ 0 h h+ ie a+ h+ ix   −˜ a 0 − h+ ie− a+ h+ ix i    −1 h+ ie a+ h+ ix   − h+ ie− a+ h+ ix



a+ e− a+ h+ ix   √ a+ e a+ h+ ix  × √ a+ e− a+ h+ ix  ˜ −h+ a ˜ − a+ h i  =  ˜ −2 a+ h+ ix 2 a+ h+ (h+ a ˜ − a+ h)e

We introduce a new coordinate y =

 +∞ x



˜ − h+ a ˜)e2 (a+ h

a+ h+ ix

˜ h+ a ˜ + a+ h

 .

r(z)dz satisfying

dy d d d = = −r(x) . dx dx dy dy Define −1 ˜ B and U (y) := A := r(x)





 a+ v + h+ wi   √

. a+ v − h+ wi e− a+ h+ ix e



a+ h+ ix

√

From Condition A, we can check that A is a smooth bounded function. The initial data U (0) = 0 follows from the fact limx→∞ (v, w)(x) = 0. Therefore, it follows from Lemma 3.12 that (v, w)(x) ≡ 0 for x ∈ [1, +∞). Going back to the original ODE system (3.105) and employing Lemma 3.12 again, one can prove v ≡ 0 from which (3.103) holds. Now, we return to the convergence of pε . By the strong convergence of θε and (3.99), one can prove that the coefficients in (3.101) satisfy Condition A. It follows from Lemma 3.11 that pε → 0 strongly in L2 (0, T0 ; L2loc (R)) as ε → 0. On the other hand, the uniform boundedness (1.42) and the interpolation theorem yield immediately that

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s pε → 0 strongly in L2 (0, T0 ; Hloc (R)), for s < s.

In the same way, we can prove (2uε − κeθ of Proposition 3.10 is completed. 2

ε

−εpε ε θx )x

→ 0 as ε → 0. Therefore the proof

Acknowledgments Feimin Huang is partially supported by National Center for Mathematics and Interdisciplinary Sciences, AMSS, CAS and NSFC Grant Nos. 11371349 and 11688101. The research of Tian-Yi Wang was supported in part by National Natural Science Foundation of China No. 11601401 and the Fundamental Research Funds for the Central Universities (WUT: 2017 IVA 072 and 2017 IVB 066). Yong Wang is partially supported by National Natural Science Foundation of China Nos. 11401565 and 11688101. References [1] T. Alazard, Incompressible limit of the nonisentropic Euler equations with the solid wall boundary conditions, Adv. Differential Equations 10 (1) (2005) 19–44. [2] T. Alazard, Low Mach number limit of the full Navier–Stokes equations, Arch. Ration. Mech. Anal. 180 (1) (2006) 1–73. [3] K. Asano, On the incompressible limit of the compressible Euler equations, Jpn. J. Appl. Math. 4 (1987) 455–488. [4] F.V. Atkinson, L.A. Peletier, Similarity solutions of the nonlinear diffusion equation, Arch. Ration. Mech. Anal. 54 (1974) 373–392. [5] G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 2000. [6] D. Bresch, B. Desjardins, E. Grenier, Oscillatory limit with changing eigen values: a formal study, in: A.V. Fursikov, G.P. Galdi, V.V. Pukhnachev (Eds.), New Directions in Mathematical Fluid Mechanics, Birkhäuser, Basel, 2010, pp. 91–105. [7] D. Bresch, B. Desjardins, E. Grenier, C.-K. Lin, Low Mach number limit of viscous polytropic flows: formal asymptotics in the periodic case, Stud. Appl. Math. 109 (2002) 125–149. [8] R. Danchin, Low Mach number limit for viscous compressible flows, ESAIM Math. Model. Numer. Anal. 39 (2005) 459–475. [9] B. Desjardins, E. Grenier, Low Mach number limit of viscous compressible flows in the whole space, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 455 (1999) 2271–2279. [10] B. Desjardins, E. Grenier, P.-L. Lions, N. Masmoudi, Incompressible limit for solutions of the isentropic Navier–Stokes equations with Dirichlet boundary conditions, J. Math. Pures Appl. 78 (1999) 461–471. [11] C. Dou, S. Jiang, Y. Ou, Low Mach number limit of full Navier–Stokes equations in a 3D bounded domain, J. Differential Equations 258 (2015) 379–398. [12] C.T. Duyn, L.A. Peletier, A class of similarity solution of the nonlinear diffusion equation, Nonlinear Anal. 1 (1977) 223–233. [13] J. Fan, H. Gao, B. Guo, Low Mach number limit of the compressible magnetohydrodynamic equations with zero thermal conductivity coefficient, Math. Methods Appl. Sci. 34 (2011) 2181–2188. [14] E. Feireisl, A. Novotny, Singular Limits in Thermodynamics of Viscous Fluids, Birkhäuser, Basel, 2009. [15] X.P. Hu, D.H. Wang, Low Mach number limit of viscous compressible magnetohydrodynamic flows, SIAM J. Math. Anal. 41 (2009) 1272–1294. [16] F.M. Huang, Thermal creep flow for the Boltzmann equation, Chin. Ann. Math. Ser. B 36 (5) (2015) 855–870. [17] F.M. Huang, A. Matsumura, Z.P. Xin, Stability of contact discontinuities for the 1-D compressible Navier–Stokes equations, Arch. Ration. Mech. Anal. 179 (2006) 55–77. [18] F.M. Huang, Y. Wang, Y. Wang, T. Yang, Justification of diffusion limit for the Boltzmann equation with a non-trivial profile, Quart. Appl. Math. 74 (4) (2016) 719–764.

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